Falkner and Boettcher, Appendix A

Percentage Accurate: 90.8% → 98.1%

Time: 9.1s

Alternatives: 16

Speedup: 1.0×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot {\left(\mathsf{fma}\left(k + 10, k, 1\right)\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 5e+274)
   (* a (* (pow k m) (pow (fma (+ k 10.0) k 1.0) -1.0)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 5e+274) {
		tmp = a * (pow(k, m) * pow(fma((k + 10.0), k, 1.0), -1.0));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 5e+274)
		tmp = Float64(a * Float64((k ^ m) * (fma(Float64(k + 10.0), k, 1.0) ^ -1.0)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+274], N[(a * N[(N[Power[k, m], $MachinePrecision] * N[Power[N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999998e274

   1. Initial program 97.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 97.3

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 97.3%

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

   if 4.9999999999999998e274 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 47.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 47.8

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in k around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   9. 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

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

   10. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot {\left(\mathsf{fma}\left(k + 10, k, 1\right)\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 5e+274)
   (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a)
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 5e+274) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 5e+274)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+274], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999998e274

   1. Initial program 97.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.3

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

      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 97.3

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

   4. Applied rewrites 97.3%

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

   if 4.9999999999999998e274 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 47.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 47.8

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in k around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   9. 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

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

   10. Applied rewrites 100.0%

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

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001, k, 0.001\right), k, 0.01\right), k, 0.1\right)}{k}, 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} - -10\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e+182)
   (/
    a
    (fma
     (* (* k k) (- 100.0 (* k k)))
     (/ (fma (fma (fma 0.0001 k 0.001) k 0.01) k 0.1) k)
     1.0))
   (if (<= m -6400000000.0)
     (/ (- a (* (/ a k) (- (/ -99.0 k) -10.0))) (* k k))
     (if (<= m 1.35)
       (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
       (* (* (* k a) k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e+182) {
		tmp = a / fma(((k * k) * (100.0 - (k * k))), (fma(fma(fma(0.0001, k, 0.001), k, 0.01), k, 0.1) / k), 1.0);
	} else if (m <= -6400000000.0) {
		tmp = (a - ((a / k) * ((-99.0 / k) - -10.0))) / (k * k);
	} else if (m <= 1.35) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e+182)
		tmp = Float64(a / fma(Float64(Float64(k * k) * Float64(100.0 - Float64(k * k))), Float64(fma(fma(fma(0.0001, k, 0.001), k, 0.01), k, 0.1) / k), 1.0));
	elseif (m <= -6400000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(Float64(-99.0 / k) - -10.0))) / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e+182], N[(a / N[(N[(N[(k * k), $MachinePrecision] * N[(100.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.0001 * k + 0.001), $MachinePrecision] * k + 0.01), $MachinePrecision] * k + 0.1), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -6400000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(N[(-99.0 / k), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.4000000000000006e182

   1. Initial program 96.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\frac{1}{10} + k \cdot \left(\frac{1}{100} + k \cdot \left(\frac{1}{1000} + \frac{1}{10000} \cdot k\right)\right)}{\color{blue}{k}}, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001, k, 0.001\right), k, 0.01\right), k, 0.1\right)}{\color{blue}{k}}, 1\right)} \]

   if -5.4000000000000006e182 < m < -6.4e9

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 77.5%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001, k, 0.001\right), k, 0.01\right), k, 0.1\right)}{k}, 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} - -10\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} - -10\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e+182)
   (/ a (fma (* (* k k) (- 100.0 (* k k))) (fma 0.001 k 0.01) 1.0))
   (if (<= m -6400000000.0)
     (/ (- a (* (/ a k) (- (/ -99.0 k) -10.0))) (* k k))
     (if (<= m 1.35)
       (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
       (* (* (* k a) k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e+182) {
		tmp = a / fma(((k * k) * (100.0 - (k * k))), fma(0.001, k, 0.01), 1.0);
	} else if (m <= -6400000000.0) {
		tmp = (a - ((a / k) * ((-99.0 / k) - -10.0))) / (k * k);
	} else if (m <= 1.35) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e+182)
		tmp = Float64(a / fma(Float64(Float64(k * k) * Float64(100.0 - Float64(k * k))), fma(0.001, k, 0.01), 1.0));
	elseif (m <= -6400000000.0)
		tmp = Float64(Float64(a - Float64(Float64(a / k) * Float64(Float64(-99.0 / k) - -10.0))) / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e+182], N[(a / N[(N[(N[(k * k), $MachinePrecision] * N[(100.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.001 * k + 0.01), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -6400000000.0], N[(N[(a - N[(N[(a / k), $MachinePrecision] * N[(N[(-99.0 / k), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.4000000000000006e182

   1. Initial program 96.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001, k, 0.01\right), k, 0.1\right)}{\color{blue}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)} \]

   if -5.4000000000000006e182 < m < -6.4e9

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 77.5%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a - \frac{a}{k} \cdot \left(\frac{-99}{k} - -10\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e+182)
   (/ a (fma (* (* k k) (- 100.0 (* k k))) (fma 0.001 k 0.01) 1.0))
   (if (<= m -6400000000.0)
     (/ a (* k k))
     (if (<= m 1.35)
       (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
       (* (* (* k a) k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e+182) {
		tmp = a / fma(((k * k) * (100.0 - (k * k))), fma(0.001, k, 0.01), 1.0);
	} else if (m <= -6400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 1.35) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e+182)
		tmp = Float64(a / fma(Float64(Float64(k * k) * Float64(100.0 - Float64(k * k))), fma(0.001, k, 0.01), 1.0));
	elseif (m <= -6400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e+182], N[(a / N[(N[(N[(k * k), $MachinePrecision] * N[(100.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.001 * k + 0.01), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -6400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.4000000000000006e182

   1. Initial program 96.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001, k, 0.01\right), k, 0.1\right)}{\color{blue}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)} \]

   if -5.4000000000000006e182 < m < -6.4e9

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 71.2%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \mathsf{fma}\left(0.001, k, 0.01\right), 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), 0.001 \cdot k, 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e+182)
   (/ a (fma (* (* k k) (- 100.0 (* k k))) (* 0.001 k) 1.0))
   (if (<= m -6400000000.0)
     (/ a (* k k))
     (if (<= m 1.35)
       (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
       (* (* (* k a) k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e+182) {
		tmp = a / fma(((k * k) * (100.0 - (k * k))), (0.001 * k), 1.0);
	} else if (m <= -6400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 1.35) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e+182)
		tmp = Float64(a / fma(Float64(Float64(k * k) * Float64(100.0 - Float64(k * k))), Float64(0.001 * k), 1.0));
	elseif (m <= -6400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e+182], N[(a / N[(N[(N[(k * k), $MachinePrecision] * N[(100.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.001 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -6400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.4000000000000006e182

   1. Initial program 96.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001, k, 0.01\right), k, 0.1\right)}{\color{blue}{k}}, 1\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), \frac{1}{1000} \cdot k, 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.8%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), 0.001 \cdot k, 1\right)} \]

   if -5.4000000000000006e182 < m < -6.4e9

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 71.2%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(100 - k \cdot k\right), 0.001 \cdot k, 1\right)}\\ \mathbf{elif}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-7} \lor \neg \left(m \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -9.2e-7) (not (<= m 7.5e-5)))
   (* (pow k m) a)
   (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -9.2e-7) || !(m <= 7.5e-5)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -9.2e-7) || !(m <= 7.5e-5))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -9.2e-7], N[Not[LessEqual[m, 7.5e-5]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -9.1999999999999998e-7 or 7.49999999999999934e-5 < m

   1. Initial program 86.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 86.0

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in k around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   9. 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

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

   10. Applied rewrites 100.0%

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

   if -9.1999999999999998e-7 < m < 7.49999999999999934e-5

   1. Initial program 93.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.3

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

      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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 92.7

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

   7. Applied rewrites 92.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-7} \lor \neg \left(m \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6400000000.0)
   (/ a (* k k))
   (if (<= m 1.35)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 1.35) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.4e9

   1. Initial program 98.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.4%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. 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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l* N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6400000000.0)
   (/ a (* k k))
   (if (<= m 1.35) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 1.35) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.4e9

   1. Initial program 98.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.4%

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

   if -6.4e9 < m < 1.3500000000000001

   1. Initial program 93.5%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 90.1%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.2e-16)
   (/ a (* k k))
   (if (<= m 1.35) (/ a (fma 10.0 k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.2e-16) {
		tmp = a / (k * k);
	} else if (m <= 1.35) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.2e-16)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.35)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.2e-16], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.2000000000000002e-16

   1. Initial program 98.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.5%

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

   if -6.2000000000000002e-16 < m < 1.3500000000000001

   1. Initial program 93.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

   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 62.8%

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

   if 1.3500000000000001 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k \cdot a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-74)
   (/ a (* k k))
   (if (<= m 0.56) (fma (fma 99.0 k -10.0) (* k a) a) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-74) {
		tmp = a / (k * k);
	} else if (m <= 0.56) {
		tmp = fma(fma(99.0, k, -10.0), (k * a), a);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e-74)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.56)
		tmp = fma(fma(99.0, k, -10.0), Float64(k * a), a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e-74], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.56], N[(N[(99.0 * k + -10.0), $MachinePrecision] * N[(k * a), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.40000000000000036e-74

   1. Initial program 98.7%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.7%

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

   if -5.40000000000000036e-74 < m < 0.56000000000000005

   1. Initial program 92.8%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 51.5%

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

   10. Applied rewrites 51.5%

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

   if 0.56000000000000005 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k \cdot a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot k, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-74)
   (/ a (* k k))
   (if (<= m 0.56) (fma (* (fma 99.0 k -10.0) k) a a) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-74) {
		tmp = a / (k * k);
	} else if (m <= 0.56) {
		tmp = fma((fma(99.0, k, -10.0) * k), a, a);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e-74)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.56)
		tmp = fma(Float64(fma(99.0, k, -10.0) * k), a, a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e-74], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.56], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.40000000000000036e-74

   1. Initial program 98.7%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.7%

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

   if -5.40000000000000036e-74 < m < 0.56000000000000005

   1. Initial program 92.8%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 51.5%

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

   10. Applied rewrites 51.5%

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

   if 0.56000000000000005 < m

   1. Initial program 76.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

      16. unpow2 N/A

          \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

      17. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left(-k, -99, -10\right), \color{blue}{k}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot k, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2e-67)
   (/ a (* k k))
   (if (<= m 0.56) (* (- a) -1.0) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-67) {
		tmp = a / (k * k);
	} else if (m <= 0.56) {
		tmp = -a * -1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-2d-67)) then
        tmp = a / (k * k)
    else if (m <= 0.56d0) then
        tmp = -a * (-1.0d0)
    else
        tmp = ((k * a) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-67) {
		tmp = a / (k * k);
	} else if (m <= 0.56) {
		tmp = -a * -1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2e-67:
		tmp = a / (k * k)
	elif m <= 0.56:
		tmp = -a * -1.0
	else:
		tmp = ((k * a) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2e-67)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.56)
		tmp = Float64(Float64(-a) * -1.0);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2e-67)
		tmp = a / (k * k);
	elseif (m <= 0.56)
		tmp = -a * -1.0;
	else
		tmp = ((k * a) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2e-67], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.56], N[((-a) * -1.0), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.99999999999999989e-67

   1. Initial program 98.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-+r+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + {k}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{{k}^{2} + \left(1 + 10 \cdot k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2} + \color{blue}{\left(10 \cdot k + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + 10 \cdot k\right) + 1}} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot k + {k}^{2}\right)} + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + \left({k}^{2} + 1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot 1\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      9. lft-mult-inverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right) \cdot k + \left({k}^{2} + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right)} + \left({k}^{2} + 1\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{a}{\left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{{k}^{2}} + \left({k}^{2} + 1\right)} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{2}\right) + 1}} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

      16. unpow2 N/A

          \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

      17. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.7%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -1.99999999999999989e-67 < m < 0.56000000000000005

   1. Initial program 92.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a \cdot {k}^{m}\right)}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot {k}^{m}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot {k}^{m}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot {k}^{m}\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-a\right)} \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      10. neg-mul-1 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{1}{\color{blue}{-1 \cdot \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}}\right) \]

      11. associate-/r* N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{\frac{1}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{\color{blue}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      13. lower-/.f64 92.9

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{-1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k}\right) \]

      16. associate-+l+ N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\mathsf{fma}\left(k + 10, k, 1\right)}\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-1 \cdot {k}^{m}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(\mathsf{neg}\left({k}^{m}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

      3. lower-pow.f64 53.5

         \[\leadsto \left(-a\right) \cdot \left(-\color{blue}{{k}^{m}}\right) \]

   7. Applied rewrites 53.5%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

   8. Taylor expanded in m around 0

      \[\leadsto \left(-a\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 51.0%

       \[\leadsto \left(-a\right) \cdot -1 \]

   if 0.56000000000000005 < m

   1. Initial program 76.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-+r+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + {k}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{{k}^{2} + \left(1 + 10 \cdot k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2} + \color{blue}{\left(10 \cdot k + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + 10 \cdot k\right) + 1}} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot k + {k}^{2}\right)} + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + \left({k}^{2} + 1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot 1\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      9. lft-mult-inverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right) \cdot k + \left({k}^{2} + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right)} + \left({k}^{2} + 1\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{a}{\left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{{k}^{2}} + \left({k}^{2} + 1\right)} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{2}\right) + 1}} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

      16. unpow2 N/A

          \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

      17. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left(-k, -99, -10\right), \color{blue}{k}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.56:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.56:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.56) (* (- a) -1.0) (* (* (* k a) k) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.56) {
		tmp = -a * -1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.56d0) then
        tmp = -a * (-1.0d0)
    else
        tmp = ((k * a) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.56) {
		tmp = -a * -1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.56:
		tmp = -a * -1.0
	else:
		tmp = ((k * a) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.56)
		tmp = Float64(Float64(-a) * -1.0);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.56)
		tmp = -a * -1.0;
	else
		tmp = ((k * a) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.56], N[((-a) * -1.0), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.56000000000000005

   1. Initial program 95.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a \cdot {k}^{m}\right)}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot {k}^{m}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot {k}^{m}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot {k}^{m}\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-a\right)} \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      10. neg-mul-1 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{1}{\color{blue}{-1 \cdot \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}}\right) \]

      11. associate-/r* N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{\frac{1}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{\color{blue}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      13. lower-/.f64 95.9

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{-1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k}\right) \]

      16. associate-+l+ N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\mathsf{fma}\left(k + 10, k, 1\right)}\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-1 \cdot {k}^{m}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(\mathsf{neg}\left({k}^{m}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

      3. lower-pow.f64 74.1

         \[\leadsto \left(-a\right) \cdot \left(-\color{blue}{{k}^{m}}\right) \]

   7. Applied rewrites 74.1%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

   8. Taylor expanded in m around 0

      \[\leadsto \left(-a\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 28.4%

       \[\leadsto \left(-a\right) \cdot -1 \]

   if 0.56000000000000005 < m

   1. Initial program 76.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-+r+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + {k}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{{k}^{2} + \left(1 + 10 \cdot k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2} + \color{blue}{\left(10 \cdot k + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + 10 \cdot k\right) + 1}} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot k + {k}^{2}\right)} + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + \left({k}^{2} + 1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot 1\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      9. lft-mult-inverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right) \cdot k + \left({k}^{2} + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right)} + \left({k}^{2} + 1\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{a}{\left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{{k}^{2}} + \left({k}^{2} + 1\right)} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{2}\right) + 1}} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

      16. unpow2 N/A

          \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

      17. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left(-k, -99, -10\right), \color{blue}{k}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.56:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 24.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.46 \cdot 10^{+16}:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.46e+16) (* (- a) -1.0) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.46e+16) {
		tmp = -a * -1.0;
	} else {
		tmp = (-10.0 * a) * 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 (m <= 1.46d+16) then
        tmp = -a * (-1.0d0)
    else
        tmp = ((-10.0d0) * a) * k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.46e+16) {
		tmp = -a * -1.0;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.46e+16:
		tmp = -a * -1.0
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.46e+16)
		tmp = Float64(Float64(-a) * -1.0);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.46e+16)
		tmp = -a * -1.0;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.46e+16], N[((-a) * -1.0), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.46e16

   1. Initial program 94.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a \cdot {k}^{m}\right)}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot {k}^{m}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot {k}^{m}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot {k}^{m}\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-a\right)} \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

      10. neg-mul-1 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{1}{\color{blue}{-1 \cdot \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}}\right) \]

      11. associate-/r* N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{\frac{1}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{\color{blue}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      13. lower-/.f64 94.7

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{-1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k}\right) \]

      16. associate-+l+ N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\mathsf{fma}\left(k + 10, k, 1\right)}\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-1 \cdot {k}^{m}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(\mathsf{neg}\left({k}^{m}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

      3. lower-pow.f64 74.5

         \[\leadsto \left(-a\right) \cdot \left(-\color{blue}{{k}^{m}}\right) \]

   7. Applied rewrites 74.5%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

   8. Taylor expanded in m around 0

      \[\leadsto \left(-a\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 28.0%

       \[\leadsto \left(-a\right) \cdot -1 \]

   if 1.46e16 < m

   1. Initial program 77.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + {k}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{{k}^{2} + \left(1 + 10 \cdot k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2} + \color{blue}{\left(10 \cdot k + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + 10 \cdot k\right) + 1}} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot k + {k}^{2}\right)} + 1} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + \left({k}^{2} + 1\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot 1\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      9. lft-mult-inverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right) \cdot k + \left({k}^{2} + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right)} \cdot k + \left({k}^{2} + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right)} + \left({k}^{2} + 1\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{a}{\left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{{k}^{2}} + \left({k}^{2} + 1\right)} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{2}\right) + 1}} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

      16. unpow2 N/A

          \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

      17. associate-*r* N/A

          \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(a \cdot k, \color{blue}{-10}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto -10 \cdot \left(a \cdot \color{blue}{k}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.7%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.46 \cdot 10^{+16}:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.4% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \left(-a\right) \cdot -1 \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* (- a) -1.0))

C

double code(double a, double k, double m) {
	return -a * -1.0;
}

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 * (-1.0d0)
end function

Java

public static double code(double a, double k, double m) {
	return -a * -1.0;
}

Python

def code(a, k, m):
	return -a * -1.0

Julia

function code(a, k, m)
	return Float64(Float64(-a) * -1.0)
end

MATLAB

function tmp = code(a, k, m)
	tmp = -a * -1.0;
end

Wolfram

code[a_, k_, m_] := N[((-a) * -1.0), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a \cdot {k}^{m}\right)}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

   3. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot {k}^{m}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot {k}^{m}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot {k}^{m}\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-a\right)} \cdot \left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left({k}^{m} \cdot \frac{1}{\mathsf{neg}\left(\left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right)}\right)} \]

   10. neg-mul-1 N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{1}{\color{blue}{-1 \cdot \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)}}\right) \]

   11. associate-/r* N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{\frac{1}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{\color{blue}{-1}}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

   13. lower-/.f64 88.4

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \color{blue}{\frac{-1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}\right) \]

   15. lift-+.f64 N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k}\right) \]

   16. associate-+l+ N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\left(-a\right) \cdot \left({k}^{m} \cdot \frac{-1}{\mathsf{fma}\left(k + 10, k, 1\right)}\right)} \]

5. Taylor expanded in k around 0

   \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-1 \cdot {k}^{m}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(\mathsf{neg}\left({k}^{m}\right)\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

   3. lower-pow.f64 83.8

      \[\leadsto \left(-a\right) \cdot \left(-\color{blue}{{k}^{m}}\right) \]

7. Applied rewrites 83.8%

   \[\leadsto \left(-a\right) \cdot \color{blue}{\left(-{k}^{m}\right)} \]

8. Taylor expanded in m around 0

   \[\leadsto \left(-a\right) \cdot -1 \]
9. Step-by-step derivation

10. Applied rewrites 19.2%

    \[\leadsto \left(-a\right) \cdot -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))