Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.7%

Time: 13.2s

Alternatives: 18

Speedup: 1.1×

• 21.8% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k}{t\_0} + \frac{10}{t\_0}, \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 2e-37)
     t_0
     (/ 1.0 (fma k (+ (/ k t_0) (/ 10.0 t_0)) (/ 1.0 t_0))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 2e-37) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(k, ((k / t_0) + (10.0 / t_0)), (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 2e-37)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(k, Float64(Float64(k / t_0) + Float64(10.0 / t_0)), Float64(1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-37], t$95$0, N[(1.0 / N[(k * N[(N[(k / t$95$0), $MachinePrecision] + N[(10.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000013e-37

   1. Initial program 95.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.00000000000000013e-37 < k

   1. Initial program 76.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 76.4

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 2: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+178}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), k, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+178)
     (/ 1.0 (fma (* (/ (pow k (- m)) a) (+ k 10.0)) k (/ 1.0 a)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+178) {
		tmp = 1.0 / fma(((pow(k, -m) / a) * (k + 10.0)), k, (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+178)
		tmp = Float64(1.0 / fma(Float64(Float64((k ^ Float64(-m)) / a) * Float64(k + 10.0)), k, Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+178], N[(1.0 / N[(N[(N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e178

   1. Initial program 96.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.6

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 80.1

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

   7. Applied rewrites 80.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 80.1

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

   9. Applied rewrites 89.2%

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

   10. Taylor expanded in m around 0

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

       1. lower-/.f64 85.4

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

   12. Applied rewrites 85.4%

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

   if 1.0000000000000001e178 < (/.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 60.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+178}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), k, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{\left(-m\right)}}{a}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-37}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_0 \cdot \left(k + 10\right), k, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (pow k (- m)) a)))
   (if (<= k 2e-37) (* a (pow k m)) (/ 1.0 (fma (* t_0 (+ k 10.0)) k t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m) / a;
	double tmp;
	if (k <= 2e-37) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / fma((t_0 * (k + 10.0)), k, t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ Float64(-m)) / a)
	tmp = 0.0
	if (k <= 2e-37)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / fma(Float64(t_0 * Float64(k + 10.0)), k, t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[k, 2e-37], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * k + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000013e-37

   1. Initial program 95.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.00000000000000013e-37 < k

   1. Initial program 76.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 76.4

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.8

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

   9. Applied rewrites 99.7%

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

Alternative 4: 20.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 0.0)
   (* k (* k (- a)))
   (fma a (* k -10.0) a)))

C

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(k * N[(k * (-a)), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 50.9

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

   5. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 30.0

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

   10. Applied rewrites 30.0%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot {k}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{k}^{2} \cdot a}\right) \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. distribute-rgt-neg-in N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. distribute-rgt-neg-in N/A

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

       10. mul-1-neg N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 13.7

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

   13. Applied rewrites 13.7%

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

   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. Initial program 72.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 35.4

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

   8. Applied rewrites 35.4%

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

4. Final simplification 20.8%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 2.7:\\ \;\;\;\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 2.7) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 2.7) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 2.7d0) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 2.7) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 2.7:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 2.7)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 2.7)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 2.7], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.7000000000000002

   1. Initial program 96.1%

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

   if 2.7000000000000002 < m

   1. Initial program 72.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.7:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.000105:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -2.1e-5)
     t_0
     (if (<= m 0.000105) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -2.1e-5) {
		tmp = t_0;
	} else if (m <= 0.000105) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -2.1e-5)
		tmp = t_0;
	elseif (m <= 0.000105)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.1e-5], t$95$0, If[LessEqual[m, 0.000105], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.09999999999999988e-5 or 1.05e-4 < m

   1. Initial program 86.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.09999999999999988e-5 < m < 1.05e-4

   1. Initial program 92.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 0.000105:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a - \frac{\mathsf{fma}\left(\frac{a}{k}, -10001, a \cdot -20\right)}{k \cdot \left(k \cdot k\right)}\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(a \cdot 99 - \frac{\mathsf{fma}\left(a, 10, \frac{a}{-k}\right)}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(\mathsf{fma}\left(a, 2000, a \cdot -6000\right), -k, a \cdot -300\right), a \cdot 20\right), -a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e+26)
   (/
    (fma
     (/ a (* k k))
     100.0
     (- a (/ (fma (/ a k) -10001.0 (* a -20.0)) (* k (* k k)))))
    (* k k))
   (if (<= m 0.48)
     (/ 1.0 (/ (fma k (+ k 10.0) 1.0) a))
     (if (<= m 1.55e+175)
       (* (* k k) (- (* a 99.0) (/ (fma a 10.0 (/ a (- k))) k)))
       (*
        (* k k)
        (fma
         k
         (fma
          k
          (fma (fma a 2000.0 (* a -6000.0)) (- k) (* a -300.0))
          (* a 20.0))
         (- a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e+26) {
		tmp = fma((a / (k * k)), 100.0, (a - (fma((a / k), -10001.0, (a * -20.0)) / (k * (k * k))))) / (k * k);
	} else if (m <= 0.48) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / a);
	} else if (m <= 1.55e+175) {
		tmp = (k * k) * ((a * 99.0) - (fma(a, 10.0, (a / -k)) / k));
	} else {
		tmp = (k * k) * fma(k, fma(k, fma(fma(a, 2000.0, (a * -6000.0)), -k, (a * -300.0)), (a * 20.0)), -a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e+26)
		tmp = Float64(fma(Float64(a / Float64(k * k)), 100.0, Float64(a - Float64(fma(Float64(a / k), -10001.0, Float64(a * -20.0)) / Float64(k * Float64(k * k))))) / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / a));
	elseif (m <= 1.55e+175)
		tmp = Float64(Float64(k * k) * Float64(Float64(a * 99.0) - Float64(fma(a, 10.0, Float64(a / Float64(-k))) / k)));
	else
		tmp = Float64(Float64(k * k) * fma(k, fma(k, fma(fma(a, 2000.0, Float64(a * -6000.0)), Float64(-k), Float64(a * -300.0)), Float64(a * 20.0)), Float64(-a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e+26], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 100.0 + N[(a - N[(N[(N[(a / k), $MachinePrecision] * -10001.0 + N[(a * -20.0), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.55e+175], N[(N[(k * k), $MachinePrecision] * N[(N[(a * 99.0), $MachinePrecision] - N[(N[(a * 10.0 + N[(a / (-k)), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * N[(N[(a * 2000.0 + N[(a * -6000.0), $MachinePrecision]), $MachinePrecision] * (-k) + N[(a * -300.0), $MachinePrecision]), $MachinePrecision] + N[(a * 20.0), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -9.2000000000000002e26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 39.7

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

   5. Applied rewrites 39.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.4

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

   10. Applied rewrites 21.4%

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

   11. Taylor expanded in k around -inf

       \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{\left(-10000 \cdot \frac{a}{k} + -1 \cdot \frac{a}{k}\right) - 20 \cdot a}{{k}^{3}}\right) - -100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{\left(-10000 \cdot \frac{a}{k} + -1 \cdot \frac{a}{k}\right) - 20 \cdot a}{{k}^{3}}\right) - -100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

   13. Applied rewrites 75.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a - \frac{\mathsf{fma}\left(\frac{a}{k}, -10001, a \cdot -20\right)}{k \cdot \left(k \cdot k\right)}\right)}{k \cdot k}} \]

   if -9.2000000000000002e26 < m < 0.47999999999999998

   1. Initial program 93.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if 0.47999999999999998 < m < 1.54999999999999992e175

   1. Initial program 66.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 23.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 32.6

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

   11. Applied rewrites 32.6%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{a}{k} + 10 \cdot a}{k} + 99 \cdot a\right)} \]

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. mul-1-neg N/A

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

       15. distribute-neg-frac2 N/A

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

       16. mul-1-neg N/A

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

       17. lower-/.f64 N/A

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

       18. mul-1-neg N/A

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

       19. lower-neg.f64 58.1

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

   14. Applied rewrites 58.1%

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

   if 1.54999999999999992e175 < m

   1. Initial program 82.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.9

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

   10. Applied rewrites 36.9%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right) + -1 \cdot a\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a, -1 \cdot a\right)} \]

   13. Applied rewrites 59.9%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(\mathsf{fma}\left(a, 2000, a \cdot -6000\right), -k, a \cdot -300\right), a \cdot 20\right), -a\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(a \cdot 99 - \frac{\mathsf{fma}\left(a, 10, \frac{a}{-k}\right)}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(\mathsf{fma}\left(a, 2000, a \cdot -6000\right), -k, a \cdot -300\right), a \cdot 20\right), -a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e+26)
   (/ (fma (/ a (* k k)) 100.0 a) (* k k))
   (if (<= m 0.48)
     (/ 1.0 (/ (fma k (+ k 10.0) 1.0) a))
     (if (<= m 1.55e+175)
       (* (* k k) (- (* a 99.0) (/ (fma a 10.0 (/ a (- k))) k)))
       (*
        (* k k)
        (fma
         k
         (fma
          k
          (fma (fma a 2000.0 (* a -6000.0)) (- k) (* a -300.0))
          (* a 20.0))
         (- a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e+26) {
		tmp = fma((a / (k * k)), 100.0, a) / (k * k);
	} else if (m <= 0.48) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / a);
	} else if (m <= 1.55e+175) {
		tmp = (k * k) * ((a * 99.0) - (fma(a, 10.0, (a / -k)) / k));
	} else {
		tmp = (k * k) * fma(k, fma(k, fma(fma(a, 2000.0, (a * -6000.0)), -k, (a * -300.0)), (a * 20.0)), -a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e+26)
		tmp = Float64(fma(Float64(a / Float64(k * k)), 100.0, a) / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / a));
	elseif (m <= 1.55e+175)
		tmp = Float64(Float64(k * k) * Float64(Float64(a * 99.0) - Float64(fma(a, 10.0, Float64(a / Float64(-k))) / k)));
	else
		tmp = Float64(Float64(k * k) * fma(k, fma(k, fma(fma(a, 2000.0, Float64(a * -6000.0)), Float64(-k), Float64(a * -300.0)), Float64(a * 20.0)), Float64(-a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e+26], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 100.0 + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.55e+175], N[(N[(k * k), $MachinePrecision] * N[(N[(a * 99.0), $MachinePrecision] - N[(N[(a * 10.0 + N[(a / (-k)), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * N[(N[(a * 2000.0 + N[(a * -6000.0), $MachinePrecision]), $MachinePrecision] * (-k) + N[(a * -300.0), $MachinePrecision]), $MachinePrecision] + N[(a * 20.0), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -9.2000000000000002e26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 39.7

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

   5. Applied rewrites 39.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.4

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

   10. Applied rewrites 21.4%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{100 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{a}{{k}^{2}} \cdot 100} + a}{{k}^{2}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{a}{{k}^{2}}, 100, a\right)}}{{k}^{2}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{{k}^{2}}}, 100, a\right)}{{k}^{2}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       8. unpow2 N/A

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

       9. lower-*.f64 73.8

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

   13. Applied rewrites 73.8%

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

   if -9.2000000000000002e26 < m < 0.47999999999999998

   1. Initial program 93.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if 0.47999999999999998 < m < 1.54999999999999992e175

   1. Initial program 66.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 23.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 32.6

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

   11. Applied rewrites 32.6%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{a}{k} + 10 \cdot a}{k} + 99 \cdot a\right)} \]

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. mul-1-neg N/A

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

       15. distribute-neg-frac2 N/A

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

       16. mul-1-neg N/A

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

       17. lower-/.f64 N/A

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

       18. mul-1-neg N/A

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

       19. lower-neg.f64 58.1

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

   14. Applied rewrites 58.1%

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

   if 1.54999999999999992e175 < m

   1. Initial program 82.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.9

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

   10. Applied rewrites 36.9%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a\right) + -1 \cdot a\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-20 \cdot \left(-100 \cdot a + 400 \cdot a\right) + 2000 \cdot a\right)\right) - \left(-100 \cdot a + 400 \cdot a\right)\right) - -20 \cdot a, -1 \cdot a\right)} \]

   13. Applied rewrites 59.9%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(\mathsf{fma}\left(a, 2000, a \cdot -6000\right), -k, a \cdot -300\right), a \cdot 20\right), -a\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{+149}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(a \cdot 99 - \frac{\mathsf{fma}\left(a, 10, \frac{a}{-k}\right)}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e+26)
   (/ (fma (/ a (* k k)) 100.0 a) (* k k))
   (if (<= m 0.48)
     (/ 1.0 (/ (fma k (+ k 10.0) 1.0) a))
     (if (<= m 6e+149)
       (* (* k k) (- (* a 99.0) (/ (fma a 10.0 (/ a (- k))) k)))
       (* k (* 99.0 (* k a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e+26) {
		tmp = fma((a / (k * k)), 100.0, a) / (k * k);
	} else if (m <= 0.48) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / a);
	} else if (m <= 6e+149) {
		tmp = (k * k) * ((a * 99.0) - (fma(a, 10.0, (a / -k)) / k));
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e+26)
		tmp = Float64(fma(Float64(a / Float64(k * k)), 100.0, a) / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / a));
	elseif (m <= 6e+149)
		tmp = Float64(Float64(k * k) * Float64(Float64(a * 99.0) - Float64(fma(a, 10.0, Float64(a / Float64(-k))) / k)));
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e+26], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 100.0 + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6e+149], N[(N[(k * k), $MachinePrecision] * N[(N[(a * 99.0), $MachinePrecision] - N[(N[(a * 10.0 + N[(a / (-k)), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -9.2000000000000002e26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 39.7

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

   5. Applied rewrites 39.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.4

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

   10. Applied rewrites 21.4%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{100 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{a}{{k}^{2}} \cdot 100} + a}{{k}^{2}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{a}{{k}^{2}}, 100, a\right)}}{{k}^{2}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{{k}^{2}}}, 100, a\right)}{{k}^{2}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       8. unpow2 N/A

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

       9. lower-*.f64 73.8

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

   13. Applied rewrites 73.8%

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

   if -9.2000000000000002e26 < m < 0.47999999999999998

   1. Initial program 93.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if 0.47999999999999998 < m < 6.00000000000000007e149

   1. Initial program 62.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 28.2

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

   11. Applied rewrites 28.2%

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

   12. Taylor expanded in k around -inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \frac{a}{k} + 10 \cdot a}{k} + 99 \cdot a\right)} \]

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. mul-1-neg N/A

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

       15. distribute-neg-frac2 N/A

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

       16. mul-1-neg N/A

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

       17. lower-/.f64 N/A

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

       18. mul-1-neg N/A

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

       19. lower-neg.f64 58.7

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

   14. Applied rewrites 58.7%

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

   if 6.00000000000000007e149 < m

   1. Initial program 82.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 29.7

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

   11. Applied rewrites 29.7%

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

   12. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

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

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. distribute-rgt-neg-in N/A

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

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 55.6

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

   14. Applied rewrites 55.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{+149}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(a \cdot 99 - \frac{\mathsf{fma}\left(a, 10, \frac{a}{-k}\right)}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e+26)
   (/ (fma (/ a (* k k)) 100.0 a) (* k k))
   (if (<= m 0.48)
     (/ 1.0 (/ (fma k (+ k 10.0) 1.0) a))
     (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e+26) {
		tmp = fma((a / (k * k)), 100.0, a) / (k * k);
	} else if (m <= 0.48) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / a);
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e+26)
		tmp = Float64(fma(Float64(a / Float64(k * k)), 100.0, a) / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / a));
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e+26], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 100.0 + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.2000000000000002e26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 39.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + k\right)} + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{1 + k \cdot \left(10 + k\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{a}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + k \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(\color{blue}{10 \cdot k} + k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right)} + k \cdot k} \]

      9. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]

      10. flip-+ N/A

          \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right) - \left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right)}{k \cdot k - \left(1 + 10 \cdot k\right)}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right) - \left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right)}{k \cdot k - \left(1 + 10 \cdot k\right)}}} \]

   7. Applied rewrites 22.6%

      \[\leadsto \frac{a}{\color{blue}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right) - \mathsf{fma}\left(k, 10, 1\right) \cdot \mathsf{fma}\left(k, 10, 1\right)}{k \cdot k - \mathsf{fma}\left(k, 10, 1\right)}}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right) - \mathsf{fma}\left(k, 10, 1\right) \cdot \mathsf{fma}\left(k, 10, 1\right)}{\color{blue}{{k}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right) - \mathsf{fma}\left(k, 10, 1\right) \cdot \mathsf{fma}\left(k, 10, 1\right)}{\color{blue}{k \cdot k}}} \]

      2. lower-*.f64 21.4

         \[\leadsto \frac{a}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right) - \mathsf{fma}\left(k, 10, 1\right) \cdot \mathsf{fma}\left(k, 10, 1\right)}{\color{blue}{k \cdot k}}} \]

   10. Applied rewrites 21.4%

       \[\leadsto \frac{a}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right) - \mathsf{fma}\left(k, 10, 1\right) \cdot \mathsf{fma}\left(k, 10, 1\right)}{\color{blue}{k \cdot k}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + 100 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{100 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{2}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{a}{{k}^{2}} \cdot 100} + a}{{k}^{2}} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{a}{{k}^{2}}, 100, a\right)}}{{k}^{2}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{{k}^{2}}}, 100, a\right)}{{k}^{2}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{\color{blue}{k \cdot k}}, 100, a\right)}{{k}^{2}} \]

       8. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{\color{blue}{k \cdot k}} \]

       9. lower-*.f64 73.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{\color{blue}{k \cdot k}} \]

   13. Applied rewrites 73.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}} \]

   if -9.2000000000000002e26 < m < 0.47999999999999998

   1. Initial program 93.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 88.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + k\right)} + 1} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      5. lower-/.f64 88.6

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)}{a}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}{a}} \]

      8. lower-+.f64 88.6

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}{a}} \]

   7. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}} \]

   if 0.47999999999999998 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 100, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e+26)
   (/ a (* k k))
   (if (<= m 0.48)
     (/ 1.0 (/ (fma k (+ k 10.0) 1.0) a))
     (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e+26) {
		tmp = a / (k * k);
	} else if (m <= 0.48) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / a);
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e+26)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / a));
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e+26], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.2000000000000002e26

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 39.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 70.0

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Applied rewrites 70.0%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -9.2000000000000002e26 < m < 0.47999999999999998

   1. Initial program 93.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 88.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + k\right)} + 1} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      5. lower-/.f64 88.6

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{a}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)}{a}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}{a}} \]

      8. lower-+.f64 88.6

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)}{a}} \]

   7. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}} \]

   if 0.47999999999999998 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.27:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.27)
   (/ a (* k k))
   (if (<= m 0.48) (/ a (fma k (+ k 10.0) 1.0)) (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.27) {
		tmp = a / (k * k);
	} else if (m <= 0.48) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.27)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.27], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.27000000000000002

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 39.5

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 67.8

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Applied rewrites 67.8%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -0.27000000000000002 < m < 0.47999999999999998

   1. Initial program 92.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 91.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   if 0.47999999999999998 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.27:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.9e-10)
   (/ a (* k k))
   (if (<= m 0.48) (/ a (fma k 10.0 1.0)) (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.9e-10) {
		tmp = a / (k * k);
	} else if (m <= 0.48) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.9e-10)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.9e-10], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.89999999999999981e-10

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 40.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 67.8

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Applied rewrites 67.8%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -2.89999999999999981e-10 < m < 0.47999999999999998

   1. Initial program 92.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 91.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{1 + 10 \cdot k}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{10 \cdot k + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot 10} + 1} \]

      3. lower-fma.f64 67.1

         \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}} \]

   8. Applied rewrites 67.1%

      \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}} \]

   if 0.47999999999999998 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.5e-11) (/ a (* k k)) (if (<= m 0.21) a (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.5e-11) {
		tmp = a / (k * k);
	} else if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-9.5d-11)) then
        tmp = a / (k * k)
    else if (m <= 0.21d0) then
        tmp = a
    else
        tmp = k * (99.0d0 * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.5e-11) {
		tmp = a / (k * k);
	} else if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -9.5e-11:
		tmp = a / (k * k)
	elif m <= 0.21:
		tmp = a
	else:
		tmp = k * (99.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.5e-11)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.21)
		tmp = a;
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -9.5e-11)
		tmp = a / (k * k);
	elseif (m <= 0.21)
		tmp = a;
	else
		tmp = k * (99.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.5e-11], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.21], a, N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.49999999999999951e-11

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 40.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 67.8

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Applied rewrites 67.8%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -9.49999999999999951e-11 < m < 0.209999999999999992

   1. Initial program 92.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 59.7

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   6. Taylor expanded in m around 0

      \[\leadsto a \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.9%

      \[\leadsto a \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 58.9

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 58.9%

       \[\leadsto \color{blue}{a} \]

   if 0.209999999999999992 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.82 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.82e-10)
   (/ a (* k 10.0))
   (if (<= m 0.21) a (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.82e-10) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.82d-10)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 0.21d0) then
        tmp = a
    else
        tmp = k * (99.0d0 * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.82e-10) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.82e-10:
		tmp = a / (k * 10.0)
	elif m <= 0.21:
		tmp = a
	else:
		tmp = k * (99.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.82e-10)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 0.21)
		tmp = a;
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.82e-10)
		tmp = a / (k * 10.0);
	elseif (m <= 0.21)
		tmp = a;
	else
		tmp = k * (99.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.82e-10], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.21], a, N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8199999999999999e-10

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 40.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{\left(k \cdot k\right)} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k\right)}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k\right)} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{1} + 1 \cdot k\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10} + 1 \cdot k\right)} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(10 + \color{blue}{k}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)}} \]

      11. lower-+.f64 48.5

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)}} \]

   8. Applied rewrites 48.5%

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(k + 10\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{10 \cdot k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

       2. lower-*.f64 28.4

          \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

   11. Applied rewrites 28.4%

       \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

   if -1.8199999999999999e-10 < m < 0.209999999999999992

   1. Initial program 92.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 59.7

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   6. Taylor expanded in m around 0

      \[\leadsto a \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.9%

      \[\leadsto a \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 58.9

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 58.9%

       \[\leadsto \color{blue}{a} \]

   if 0.209999999999999992 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.82 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.21) a (* k (* 99.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.21d0) then
        tmp = a
    else
        tmp = k * (99.0d0 * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = k * (99.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.21:
		tmp = a
	else:
		tmp = k * (99.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.21)
		tmp = a;
	else
		tmp = Float64(k * Float64(99.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.21)
		tmp = a;
	else
		tmp = k * (99.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.21], a, N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.209999999999999992

   1. Initial program 96.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 77.7

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   6. Taylor expanded in m around 0

      \[\leadsto a \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.0%

      \[\leadsto a \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 33.0

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 33.0%

       \[\leadsto \color{blue}{a} \]

   if 0.209999999999999992 < m

   1. Initial program 72.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 17.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), 99 \cdot a\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + 99 \cdot \left(a \cdot k\right)}, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot \left(a \cdot k\right) + -10 \cdot a}, a\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(99 \cdot a\right) \cdot k} + -10 \cdot a, a\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot \left(99 \cdot a\right)} + -10 \cdot a, a\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99 \cdot a, -10 \cdot a\right)}, a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, \color{blue}{a \cdot 99}, -10 \cdot a\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

       8. lower-*.f64 29.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot 99, \color{blue}{a \cdot -10}\right), a\right) \]

   11. Applied rewrites 29.0%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   12. Taylor expanded in k around inf

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-99\right)\right)} \cdot \left(a \cdot k\right)\right) \]

       6. distribute-lft-neg-in N/A

          \[\leadsto k \cdot \color{blue}{\left(\mathsf{neg}\left(-99 \cdot \left(a \cdot k\right)\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-99 \cdot a\right) \cdot k}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot -99\right)} \cdot k\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(-99 \cdot k\right)}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{k \cdot \left(\mathsf{neg}\left(a \cdot \left(-99 \cdot k\right)\right)\right)} \]

       11. *-commutative N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(k \cdot -99\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto k \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot k\right) \cdot -99}\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot \left(\mathsf{neg}\left(-99\right)\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto k \cdot \left(\left(a \cdot k\right) \cdot \color{blue}{99}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto k \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 99\right)} \]

       16. *-commutative N/A

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

       17. lower-*.f64 51.5

           \[\leadsto k \cdot \left(\color{blue}{\left(k \cdot a\right)} \cdot 99\right) \]

   14. Applied rewrites 51.5%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot a\right) \cdot 99\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 21.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, k \cdot -10, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma a (* k -10.0) a))

C

double code(double a, double k, double m) {
	return fma(a, (k * -10.0), a);
}

Julia

function code(a, k, m)
	return fma(a, Float64(k * -10.0), a)
end

Wolfram

code[a_, k_, m_] := N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 88.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 46.2

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Applied rewrites 46.2%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]

   5. lower-*.f64 23.9

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

8. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]
9. Add Preprocessing

Alternative 18: 20.4% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return 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 = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 88.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   2. lower-pow.f64 85.1

      \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

5. Applied rewrites 85.1%

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

6. Taylor expanded in m around 0

   \[\leadsto a \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 23.3%

   \[\leadsto a \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 23.3

      \[\leadsto \color{blue}{a} \]

10. Applied rewrites 23.3%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))