Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.3%

Time: 9.0s

Alternatives: 15

Speedup: 0.5×

• 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e-31)
   (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a)
   (if (<= m 9.2e-9)
     (pow (fma (+ (/ k a) (/ 10.0 a)) k (pow a -1.0)) -1.0)
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e-31) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else if (m <= 9.2e-9) {
		tmp = pow(fma(((k / a) + (10.0 / a)), k, pow(a, -1.0)), -1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e-31)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	elseif (m <= 9.2e-9)
		tmp = fma(Float64(Float64(k / a) + Float64(10.0 / a)), k, (a ^ -1.0)) ^ -1.0;
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e-31], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 9.2e-9], N[Power[N[(N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision] * k + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.1999999999999994e-31

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -9.1999999999999994e-31 < m < 9.1999999999999997e-9

   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. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.4

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 99.0%

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

   if 9.1999999999999997e-9 < m

   1. Initial program 77.2%

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

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

      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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{10}{t\_0} + \frac{k}{t\_0}, k, {t\_0}^{-1}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 3.5e-10)
     (* (fma -10.0 k 1.0) t_0)
     (pow (fma (+ (/ 10.0 t_0) (/ k t_0)) k (pow t_0 -1.0)) -1.0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 3.5e-10) {
		tmp = fma(-10.0, k, 1.0) * t_0;
	} else {
		tmp = pow(fma(((10.0 / t_0) + (k / t_0)), k, pow(t_0, -1.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 3.5e-10)
		tmp = Float64(fma(-10.0, k, 1.0) * t_0);
	else
		tmp = fma(Float64(Float64(10.0 / t_0) + Float64(k / t_0)), k, (t_0 ^ -1.0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 3.5e-10], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[Power[N[(N[(N[(10.0 / t$95$0), $MachinePrecision] + N[(k / t$95$0), $MachinePrecision]), $MachinePrecision] * k + N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999998e-10

   1. Initial program 95.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

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

      8. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. exp-prod N/A

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

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

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   if 3.4999999999999998e-10 < k

   1. Initial program 82.8%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 82.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 82.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{10}{{k}^{m} \cdot a} + \frac{k}{{k}^{m} \cdot a}, k, {\left({k}^{m} \cdot a\right)}^{-1}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-5} \lor \neg \left(m \leq 9.2 \cdot 10^{-9}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -7.2e-5) (not (<= m 9.2e-9)))
   (* (pow k m) a)
   (pow (fma (+ (/ k a) (/ 10.0 a)) k (pow a -1.0)) -1.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -7.2e-5) || !(m <= 9.2e-9)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = pow(fma(((k / a) + (10.0 / a)), k, pow(a, -1.0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -7.2e-5) || !(m <= 9.2e-9))
		tmp = Float64((k ^ m) * a);
	else
		tmp = fma(Float64(Float64(k / a) + Float64(10.0 / a)), k, (a ^ -1.0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -7.2e-5], N[Not[LessEqual[m, 9.2e-9]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[Power[N[(N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision] * k + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -7.20000000000000018e-5 or 9.1999999999999997e-9 < m

   1. Initial program 87.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.9

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

      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 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if -7.20000000000000018e-5 < m < 9.1999999999999997e-9

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.7

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-5} \lor \neg \left(m \leq 9.2 \cdot 10^{-9}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.5e-5)
   (/ (* a (pow k m)) (fma 10.0 k 1.0))
   (if (<= m 9.2e-9)
     (pow (fma (+ (/ k a) (/ 10.0 a)) k (pow a -1.0)) -1.0)
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.5e-5) {
		tmp = (a * pow(k, m)) / fma(10.0, k, 1.0);
	} else if (m <= 9.2e-9) {
		tmp = pow(fma(((k / a) + (10.0 / a)), k, pow(a, -1.0)), -1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.5e-5)
		tmp = Float64(Float64(a * (k ^ m)) / fma(10.0, k, 1.0));
	elseif (m <= 9.2e-9)
		tmp = fma(Float64(Float64(k / a) + Float64(10.0 / a)), k, (a ^ -1.0)) ^ -1.0;
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.5e-5], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9.2e-9], N[Power[N[(N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision] * k + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.49999999999999943e-5

   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 k around 0

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + 10 \cdot k}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -6.49999999999999943e-5 < m < 9.1999999999999997e-9

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.7

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.6%

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

   if 9.1999999999999997e-9 < m

   1. Initial program 77.2%

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

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

      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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.26:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.26)
   (/ (/ (* (/ (/ a k) k) 99.0) k) k)
   (if (<= m 1.0)
     (pow (fma (+ (/ k a) (/ 10.0 a)) k (pow a -1.0)) -1.0)
     (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.26) {
		tmp = ((((a / k) / k) * 99.0) / k) / k;
	} else if (m <= 1.0) {
		tmp = pow(fma(((k / a) + (10.0 / a)), k, pow(a, -1.0)), -1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.26)
		tmp = Float64(Float64(Float64(Float64(Float64(a / k) / k) * 99.0) / k) / k);
	elseif (m <= 1.0)
		tmp = fma(Float64(Float64(k / a) + Float64(10.0 / a)), k, (a ^ -1.0)) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.26], N[(N[(N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 1.0], N[Power[N[(N[(N[(k / a), $MachinePrecision] + N[(10.0 / a), $MachinePrecision]), $MachinePrecision] * k + N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.26000000000000001

   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 28.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{99 \cdot \frac{a}{{k}^{2}}}{k}}{k} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.1%

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

   if -0.26000000000000001 < m < 1

   1. Initial program 94.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 96.1%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.26:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{k}{a} + \frac{10}{a}, k, {a}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.65:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(k, k, -100\right)}{k - 10}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.65)
   (/ (/ (* (/ (/ a k) k) 99.0) k) k)
   (if (<= m 1.0)
     (/ a (fma (/ (fma k k -100.0) (- k 10.0)) k 1.0))
     (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.65) {
		tmp = ((((a / k) / k) * 99.0) / k) / k;
	} else if (m <= 1.0) {
		tmp = a / fma((fma(k, k, -100.0) / (k - 10.0)), k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.65)
		tmp = Float64(Float64(Float64(Float64(Float64(a / k) / k) * 99.0) / k) / k);
	elseif (m <= 1.0)
		tmp = Float64(a / fma(Float64(fma(k, k, -100.0) / Float64(k - 10.0)), k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.65], N[(N[(N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(N[(N[(k * k + -100.0), $MachinePrecision] / N[(k - 10.0), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.650000000000000022

   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 28.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{99 \cdot \frac{a}{{k}^{2}}}{k}}{k} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.1%

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

   if -0.650000000000000022 < m < 1

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 7: 71.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.92:\\ \;\;\;\;\frac{a - \frac{10 - \frac{99}{k}}{k} \cdot a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(k, k, -100\right)}{k - 10}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.92)
   (/ (- a (* (/ (- 10.0 (/ 99.0 k)) k) a)) (* k k))
   (if (<= m 1.0)
     (/ a (fma (/ (fma k k -100.0) (- k 10.0)) k 1.0))
     (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.92) {
		tmp = (a - (((10.0 - (99.0 / k)) / k) * a)) / (k * k);
	} else if (m <= 1.0) {
		tmp = a / fma((fma(k, k, -100.0) / (k - 10.0)), k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.92)
		tmp = Float64(Float64(a - Float64(Float64(Float64(10.0 - Float64(99.0 / k)) / k) * a)) / Float64(k * k));
	elseif (m <= 1.0)
		tmp = Float64(a / fma(Float64(fma(k, k, -100.0) / Float64(k - 10.0)), k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.92], N[(N[(a - N[(N[(N[(10.0 - N[(99.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(N[(N[(k * k + -100.0), $MachinePrecision] / N[(k - 10.0), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.92000000000000004

   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 28.3%

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

   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 3.4%

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

   9. Taylor expanded in k around -inf

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

   11. Applied rewrites 70.6%

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

   if -0.92000000000000004 < m < 1

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 8: 69.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.78:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(k, k, -100\right)}{k - 10}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.78)
   (/ a (* k k))
   (if (<= m 1.0)
     (/ a (fma (/ (fma k k -100.0) (- k 10.0)) k 1.0))
     (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.78) {
		tmp = a / (k * k);
	} else if (m <= 1.0) {
		tmp = a / fma((fma(k, k, -100.0) / (k - 10.0)), k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.78)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.0)
		tmp = Float64(a / fma(Float64(fma(k, k, -100.0) / Float64(k - 10.0)), k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.78], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(N[(N[(k * k + -100.0), $MachinePrecision] / N[(k - 10.0), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.78000000000000003

   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 28.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.0%

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

   if -0.78000000000000003 < m < 1

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 9: 53.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -7.5 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -5.8e-98)
     t_0
     (if (<= m -7.5e-281)
       (fma (* (fma 99.0 k -10.0) a) k a)
       (if (<= m 0.92) t_0 (* (* (* a k) k) 99.0))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -5.8e-98) {
		tmp = t_0;
	} else if (m <= -7.5e-281) {
		tmp = fma((fma(99.0, k, -10.0) * a), k, a);
	} else if (m <= 0.92) {
		tmp = t_0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -5.8e-98)
		tmp = t_0;
	elseif (m <= -7.5e-281)
		tmp = fma(Float64(fma(99.0, k, -10.0) * a), k, a);
	elseif (m <= 0.92)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.8e-98], t$95$0, If[LessEqual[m, -7.5e-281], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a), $MachinePrecision] * k + a), $MachinePrecision], If[LessEqual[m, 0.92], t$95$0, N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.8e-98 or -7.49999999999999968e-281 < m < 0.92000000000000004

   1. Initial program 98.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 61.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.7%

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

   if -5.8e-98 < m < -7.49999999999999968e-281

   1. Initial program 89.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 89.0%

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

   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 59.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 59.5%

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

   if 0.92000000000000004 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 10: 53.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -7.5 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -5.8e-98)
     t_0
     (if (<= m -7.5e-281)
       (fma (* a k) -10.0 a)
       (if (<= m 0.92) t_0 (* (* (* a k) k) 99.0))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -5.8e-98) {
		tmp = t_0;
	} else if (m <= -7.5e-281) {
		tmp = fma((a * k), -10.0, a);
	} else if (m <= 0.92) {
		tmp = t_0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -5.8e-98)
		tmp = t_0;
	elseif (m <= -7.5e-281)
		tmp = fma(Float64(a * k), -10.0, a);
	elseif (m <= 0.92)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.8e-98], t$95$0, If[LessEqual[m, -7.5e-281], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision], If[LessEqual[m, 0.92], t$95$0, N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.8e-98 or -7.49999999999999968e-281 < m < 0.92000000000000004

   1. Initial program 98.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 61.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.7%

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

   if -5.8e-98 < m < -7.49999999999999968e-281

   1. Initial program 89.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 89.0%

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

   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 57.7%

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

   if 0.92000000000000004 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 11: 69.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.78:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.78)
   (/ a (* k k))
   (if (<= m 1.0) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.78) {
		tmp = a / (k * k);
	} else if (m <= 1.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.78)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.78], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.78000000000000003

   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 28.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.0%

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

   if -0.78000000000000003 < m < 1

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.0%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 12: 58.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e-97)
   (/ a (* k k))
   (if (<= m 1.0) (/ a (fma 10.0 k 1.0)) (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e-97) {
		tmp = a / (k * k);
	} else if (m <= 1.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e-97)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e-97], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.4000000000000001e-97

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 36.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.9%

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

   if -1.4000000000000001e-97 < m < 1

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.0%

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

   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 63.5%

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

   if 1 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

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

Alternative 13: 36.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.098:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.098) (* 1.0 a) (* (* (* a k) k) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.098) {
		tmp = 1.0 * a;
	} else {
		tmp = ((a * k) * 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 <= 0.098d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((a * k) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.098) {
		tmp = 1.0 * a;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.098:
		tmp = 1.0 * a
	else:
		tmp = ((a * k) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.098)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.098)
		tmp = 1.0 * a;
	else
		tmp = ((a * k) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.098], N[(1.0 * a), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.098000000000000004

   1. Initial program 96.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.3

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

      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 96.3

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

   4. Applied rewrites 96.3%

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

   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. lower-fma.f64 N/A

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

      5. lower-+.f64 66.9

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 28.1%

       \[\leadsto 1 \cdot a \]

   if 0.098000000000000004 < m

   1. Initial program 77.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.0%

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

   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 24.6%

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

   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 52.1%

       \[\leadsto \left(\left(a \cdot k\right) \cdot k\right) \cdot 99 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 8.2e+29) (* 1.0 a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.2e+29) {
		tmp = 1.0 * a;
	} 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 <= 8.2d+29) then
        tmp = 1.0d0 * a
    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 <= 8.2e+29) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.2e+29:
		tmp = 1.0 * a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.2e+29)
		tmp = Float64(1.0 * a);
	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 <= 8.2e+29)
		tmp = 1.0 * a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.2e+29], N[(1.0 * a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.2000000000000007e29

   1. Initial program 95.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.2

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

      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 95.2

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

   4. Applied rewrites 95.2%

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

   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. lower-fma.f64 N/A

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

      5. lower-+.f64 65.8

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 27.6%

       \[\leadsto 1 \cdot a \]

   if 8.2000000000000007e29 < m

   1. Initial program 78.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 3.1%

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

   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 8.2%

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

   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 23.5%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 20.4% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* 1.0 a))

C

double code(double a, double k, double m) {
	return 1.0 * a;
}

Fortran

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

Java

public static double code(double a, double k, double m) {
	return 1.0 * a;
}

Python

def code(a, k, m):
	return 1.0 * a

Julia

function code(a, k, m)
	return Float64(1.0 * a)
end

MATLAB

function tmp = code(a, k, m)
	tmp = 1.0 * a;
end

Wolfram

code[a_, k_, m_] := N[(1.0 * a), $MachinePrecision]

Derivation

1. Initial program 90.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. 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 90.4

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

   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 90.4

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

4. Applied rewrites 90.4%

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

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. lower-fma.f64 N/A

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

   5. lower-+.f64 47.2

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

7. Applied rewrites 47.2%

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

8. Taylor expanded in k around 0

   \[\leadsto 1 \cdot a \]
9. Step-by-step derivation

10. Applied rewrites 20.5%

    \[\leadsto 1 \cdot a \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))