Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.6%

Time: 9.0s

Alternatives: 14

Speedup: 1.1×

• 22.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 3.0) (/ t_0 (fma k 10.0 (fma k k 1.0))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 3.0) {
		tmp = t_0 / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 3.0)
		tmp = Float64(t_0 / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, 3.0], N[(t$95$0 / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 98.2

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

   4. Applied rewrites 98.2%

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

   if 3 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{a} + \frac{\frac{10}{k}}{a}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{a - \frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ 1.0 (* (* (+ (/ 1.0 a) (/ (/ 10.0 k) a)) k) k))
     (if (<= t_0 1e+301)
       (/ a (fma (/ 1.0 (/ 1.0 (+ 10.0 k))) k 1.0))
       (if (<= t_0 INFINITY)
         (/ (/ (- a (/ (fma -99.0 (/ a k) (* 10.0 a)) k)) k) k)
         (fma (fma -10.0 a (* (* k a) 99.0)) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = (pow(k, m) * a) / (((10.0 * k) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / ((((1.0 / a) + ((10.0 / k) / a)) * k) * k);
	} else if (t_0 <= 1e+301) {
		tmp = a / fma((1.0 / (1.0 / (10.0 + k))), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((a - (fma(-99.0, (a / k), (10.0 * a)) / k)) / k) / k;
	} else {
		tmp = fma(fma(-10.0, a, ((k * a) * 99.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64((k ^ m) * a) / Float64(Float64(Float64(10.0 * k) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(1.0 / a) + Float64(Float64(10.0 / k) / a)) * k) * k));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / fma(Float64(1.0 / Float64(1.0 / Float64(10.0 + k))), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(a - Float64(fma(-99.0, Float64(a / k), Float64(10.0 * a)) / k)) / k) / k);
	else
		tmp = fma(fma(-10.0, a, Float64(Float64(k * a) * 99.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(N[(10.0 / k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(1.0 / N[(1.0 / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(a - N[(N[(-99.0 * N[(a / k), $MachinePrecision] + N[(10.0 * a), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], N[(N[(-10.0 * a + N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 97.9

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

      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 97.9

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

      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 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 44.7%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 97.6%

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

   7. Applied rewrites 97.6%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 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 49.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{a} + \frac{\frac{10}{k}}{a}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\frac{a - \frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{a} + \frac{\frac{10}{k}}{a}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ 1.0 (* (* (+ (/ 1.0 a) (/ (/ 10.0 k) a)) k) k))
     (if (<= t_0 1e+301)
       (/ a (fma (/ 1.0 (/ 1.0 (+ 10.0 k))) k 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma (fma -10.0 a (* (* k a) 99.0)) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = (pow(k, m) * a) / (((10.0 * k) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / ((((1.0 / a) + ((10.0 / k) / a)) * k) * k);
	} else if (t_0 <= 1e+301) {
		tmp = a / fma((1.0 / (1.0 / (10.0 + k))), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(fma(-10.0, a, ((k * a) * 99.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64((k ^ m) * a) / Float64(Float64(Float64(10.0 * k) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(1.0 / a) + Float64(Float64(10.0 / k) / a)) * k) * k));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / fma(Float64(1.0 / Float64(1.0 / Float64(10.0 + k))), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(fma(-10.0, a, Float64(Float64(k * a) * 99.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(N[(10.0 / k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(1.0 / N[(1.0 / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a + N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 97.9

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

      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 97.9

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

      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 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 44.7%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 97.6%

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

   7. Applied rewrites 97.6%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 inf

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

   8. Applied rewrites 41.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{1}{\left(\left(\frac{1}{a} + \frac{\frac{10}{k}}{a}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{-1}{10} \cdot \mathsf{fma}\left(k, k, -100\right), k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (fma (* (/ -1.0 10.0) (fma k k -100.0)) k 1.0))
     (if (<= t_0 1e+301)
       (/ a (fma (/ 1.0 (/ 1.0 (+ 10.0 k))) k 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma (fma -10.0 a (* (* k a) 99.0)) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = (pow(k, m) * a) / (((10.0 * k) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(((-1.0 / 10.0) * fma(k, k, -100.0)), k, 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / fma((1.0 / (1.0 / (10.0 + k))), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(fma(-10.0, a, ((k * a) * 99.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64((k ^ m) * a) / Float64(Float64(Float64(10.0 * k) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(Float64(Float64(-1.0 / 10.0) * fma(k, k, -100.0)), k, 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / fma(Float64(1.0 / Float64(1.0 / Float64(10.0 + k))), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(fma(-10.0, a, Float64(Float64(k * a) * 99.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(-1.0 / 10.0), $MachinePrecision] * N[(k * k + -100.0), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(1.0 / N[(1.0 / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a + N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 48.0%

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 48.4%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 97.6%

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

   7. Applied rewrites 97.6%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 inf

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

   8. Applied rewrites 41.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{-1}{10} \cdot \mathsf{fma}\left(k, k, -100\right), k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{-1}{10} \cdot \mathsf{fma}\left(k, k, -100\right), k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (fma (* (/ -1.0 10.0) (fma k k -100.0)) k 1.0))
     (if (<= t_0 1e+301)
       (/ a (fma (+ 10.0 k) k 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma (fma -10.0 a (* (* k a) 99.0)) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = (pow(k, m) * a) / (((10.0 * k) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(((-1.0 / 10.0) * fma(k, k, -100.0)), k, 1.0);
	} else if (t_0 <= 1e+301) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(fma(-10.0, a, ((k * a) * 99.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64((k ^ m) * a) / Float64(Float64(Float64(10.0 * k) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(Float64(Float64(-1.0 / 10.0) * fma(k, k, -100.0)), k, 1.0));
	elseif (t_0 <= 1e+301)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(fma(-10.0, a, Float64(Float64(k * a) * 99.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(-1.0 / 10.0), $MachinePrecision] * N[(k * k + -100.0), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a + N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 48.0%

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 48.4%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 97.6%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 inf

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

   8. Applied rewrites 41.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{-1}{10} \cdot \mathsf{fma}\left(k, k, -100\right), k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 10^{+301}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.0) (* (/ (pow k m) (fma (+ 10.0 k) k 1.0)) a) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.0) {
		tmp = (pow(k, m) / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.0)
		tmp = Float64(Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.0], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

   if 3 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.00036:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -5.3e-5)
     (/ t_0 (* k k))
     (if (<= m 0.00036) (/ a (fma (/ 1.0 (/ 1.0 (+ 10.0 k))) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -5.3e-5) {
		tmp = t_0 / (k * k);
	} else if (m <= 0.00036) {
		tmp = a / fma((1.0 / (1.0 / (10.0 + k))), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -5.3e-5)
		tmp = Float64(t_0 / Float64(k * k));
	elseif (m <= 0.00036)
		tmp = Float64(a / fma(Float64(1.0 / Float64(1.0 / Float64(10.0 + k))), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -5.3e-5], N[(t$95$0 / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00036], N[(a / N[(N[(1.0 / N[(1.0 / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.3000000000000001e-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 inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -5.3000000000000001e-5 < m < 3.60000000000000023e-4

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 96.6%

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

   7. Applied rewrites 96.6%

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

   if 3.60000000000000023e-4 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.00036:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -28:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.00036:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{1}{\frac{1}{10 + k}}, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -28.0)
     t_0
     (if (<= m 0.00036) (/ a (fma (/ 1.0 (/ 1.0 (+ 10.0 k))) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -28.0) {
		tmp = t_0;
	} else if (m <= 0.00036) {
		tmp = a / fma((1.0 / (1.0 / (10.0 + k))), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -28.0)
		tmp = t_0;
	elseif (m <= 0.00036)
		tmp = Float64(a / fma(Float64(1.0 / Float64(1.0 / Float64(10.0 + k))), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -28.0], t$95$0, If[LessEqual[m, 0.00036], N[(a / N[(N[(1.0 / N[(1.0 / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -28 or 3.60000000000000023e-4 < m

   1. Initial program 84.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -28 < m < 3.60000000000000023e-4

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 95.9%

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

   7. Applied rewrites 95.9%

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

Alternative 9: 60.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -28:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.5:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -28.0)
   (/ a (* k k))
   (if (<= m 2.5)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (fma -10.0 a (* (* k a) 99.0)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -28.0) {
		tmp = a / (k * k);
	} else if (m <= 2.5) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(-10.0, a, ((k * a) * 99.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -28.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.5)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(fma(-10.0, a, Float64(Float64(k * a) * 99.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -28.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.5], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a + N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -28

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.6%

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

   if -28 < m < 2.5

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 95.9%

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

   if 2.5 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 29.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -28:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.5:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-10, a, \left(k \cdot a\right) \cdot 99\right), k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -28:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -28.0)
   (/ a (* k k))
   (if (<= m 0.65) (/ a (fma (+ 10.0 k) k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -28.0) {
		tmp = a / (k * k);
	} else if (m <= 0.65) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -28.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.65)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -28.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.65], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -28

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 63.6%

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

   if -28 < m < 0.650000000000000022

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 95.9%

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

   if 0.650000000000000022 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   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 22.8%

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

Alternative 11: 47.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-46)
   (/ a (* k k))
   (if (<= m 0.65) (/ a (fma 10.0 k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-46) {
		tmp = a / (k * k);
	} else if (m <= 0.65) {
		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 <= -5.4e-46)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.65)
		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, -5.4e-46], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.65], 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 < -5.4e-46

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 64.6%

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

   if -5.4e-46 < m < 0.650000000000000022

   1. Initial program 96.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 96.4%

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

   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 66.0%

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

   if 0.650000000000000022 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   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 22.8%

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

Alternative 12: 46.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -5.4e-301) t_0 (if (<= k 0.1) (fma (* -10.0 a) k a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -5.4e-301) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -5.4e-301)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -5.4e-301], t$95$0, If[LessEqual[k, 0.1], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -5.3999999999999999e-301 or 0.10000000000000001 < k

   1. Initial program 83.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 45.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 48.1%

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

   if -5.3999999999999999e-301 < k < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 51.7%

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

   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 50.8%

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

   10. Applied rewrites 50.8%

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

Alternative 13: 25.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.55:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.55) (* 1.0 a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.55) {
		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 <= 0.55d0) 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 <= 0.55) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.55:
		tmp = 1.0 * a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.55)
		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 <= 0.55)
		tmp = 1.0 * a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.55], N[(1.0 * a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.55000000000000004

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.2

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 29.5%

       \[\leadsto 1 \cdot a \]

   if 0.55000000000000004 < m

   1. Initial program 72.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   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 22.8%

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

Alternative 14: 19.9% accurate, 22.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

   1. lift-/.f64 N/A

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

   2. 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 89.1

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

   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 89.1

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

4. Applied rewrites 89.1%

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

5. Taylor expanded in k around 0

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

   1. lower-pow.f64 81.5

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

7. Applied rewrites 81.5%

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

8. Taylor expanded in m around 0

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

10. Applied rewrites 20.4%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))