Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 98.0%

Time: 8.3s

Alternatives: 10

Speedup: 1.1×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.6%

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

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

   1. Initial program 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
   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 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(1 + \color{blue}{k \cdot \left(99 \cdot k - 10\right)}\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. 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 98.6

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

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

   1. Initial program 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
   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 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(1 + \color{blue}{k \cdot \left(99 \cdot k - 10\right)}\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-16} \lor \neg \left(m \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.5e-16) (not (<= m 1.4e-5)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.5e-16) || !(m <= 1.4e-5)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3.5e-16) || !(m <= 1.4e-5))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3.5e-16], N[Not[LessEqual[m, 1.4e-5]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.50000000000000017e-16 or 1.39999999999999998e-5 < m

   1. Initial program 88.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.0

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

      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 88.0

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

   4. Applied rewrites 88.0%

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

   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%

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

   if -3.50000000000000017e-16 < m < 1.39999999999999998e-5

   1. Initial program 95.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-16} \lor \neg \left(m \leq 1.4 \cdot 10^{-5}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;\frac{\frac{99}{k \cdot k}}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.00026:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.24)
   (* (/ (/ 99.0 (* k k)) (* k k)) a)
   (if (<= m 0.00026)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.24) {
		tmp = ((99.0 / (k * k)) / (k * k)) * a;
	} else if (m <= 0.00026) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.24)
		tmp = Float64(Float64(Float64(99.0 / Float64(k * k)) / Float64(k * k)) * a);
	elseif (m <= 0.00026)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.24], N[(N[(N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.00026], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.23999999999999999

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
   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 45.4

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

   7. Applied rewrites 45.4%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{\left(1 + \frac{99}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 69.0%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 77.1%

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

   if -0.23999999999999999 < m < 2.59999999999999977e-4

   1. Initial program 96.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.3%

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

   if 2.59999999999999977e-4 < m

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. 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 76.7

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

      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 76.7

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

   4. Applied rewrites 76.7%

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

   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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(1 + \color{blue}{k \cdot \left(99 \cdot k - 10\right)}\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 33.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.00026:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.29)
   (/ a (* k k))
   (if (<= m 0.00026)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.29) {
		tmp = a / (k * k);
	} else if (m <= 0.00026) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.29)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.00026)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.29], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00026], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.28999999999999998

   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 45.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.0%

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

   if -0.28999999999999998 < m < 2.59999999999999977e-4

   1. Initial program 96.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.3%

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

   if 2.59999999999999977e-4 < m

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. 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 76.7

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

      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 76.7

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

   4. Applied rewrites 76.7%

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

   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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(1 + \color{blue}{k \cdot \left(99 \cdot k - 10\right)}\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 33.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.00026:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.6e-7)
   (/ a (* k k))
   (if (<= m 0.00026)
     (/ a (fma 10.0 k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.6e-7) {
		tmp = a / (k * k);
	} else if (m <= 0.00026) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.6e-7)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.00026)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.6e-7], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00026], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.59999999999999999e-7

   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 45.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.0%

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

   if -2.59999999999999999e-7 < m < 2.59999999999999977e-4

   1. Initial program 96.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.3%

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

   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 67.6%

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

   if 2.59999999999999977e-4 < m

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. 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 76.7

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

      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 76.7

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

   4. Applied rewrites 76.7%

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

   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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(1 + \color{blue}{k \cdot \left(99 \cdot k - 10\right)}\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 33.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 48.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2150000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.6e-7)
   (/ a (* k k))
   (if (<= m 2150000000000.0) (/ a (fma 10.0 k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.6e-7) {
		tmp = a / (k * k);
	} else if (m <= 2150000000000.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.6e-7)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2150000000000.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.6e-7], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2150000000000.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.59999999999999999e-7

   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 45.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.0%

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

   if -2.59999999999999999e-7 < m < 2.15e12

   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 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 92.3%

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

   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 65.5%

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

   if 2.15e12 < m

   1. Initial program 77.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 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 7.4%

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

   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 20.2%

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

Alternative 8: 45.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -9.5 \cdot 10^{-138} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -9.5e-138) (not (<= k 0.1)))
   (/ a (* k k))
   (fma (* a k) -10.0 a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9.5e-138) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = fma((a * k), -10.0, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -9.5e-138) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(a * k), -10.0, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -9.5e-138], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -9.49999999999999997e-138 or 0.10000000000000001 < k

   1. Initial program 83.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 48.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 49.0%

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

   if -9.49999999999999997e-138 < k < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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 44.6%

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

   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 44.3%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.5 \cdot 10^{-138} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2150000000000:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2150000000000.0) (* 1.0 a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2150000000000.0) {
		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 <= 2150000000000.0d0) 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 <= 2150000000000.0) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2150000000000.0:
		tmp = 1.0 * a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2150000000000.0)
		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 <= 2150000000000.0)
		tmp = 1.0 * a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2150000000000.0], N[(1.0 * a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.15e12

   1. Initial program 97.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.4

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

      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 97.4

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

   4. Applied rewrites 97.4%

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

   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 69.2

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 28.9%

       \[\leadsto 1 \cdot a \]

   if 2.15e12 < m

   1. Initial program 77.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 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 7.4%

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

   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 20.2%

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

Alternative 10: 20.2% 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 90.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 90.5

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

   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 90.5

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

4. Applied rewrites 90.5%

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

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 46.7

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

7. Applied rewrites 46.7%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 20.3%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))