Falkner and Boettcher, Appendix A

Percentage Accurate: 90.7% → 98.4%

Time: 9.5s

Alternatives: 14

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 3866.8335450524555
  2. k = -1.5589539246936062e+264
  3. m = -2.0105721723568538e+149

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{+156}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -5e+156)
   (* a (pow k m))
   (if (<= k 1.2e+154)
     (* (/ (pow k m) (fma (+ 10.0 k) k 1.0)) a)
     (* (pow k (+ -1.0 m)) (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e+156) {
		tmp = a * pow(k, m);
	} else if (k <= 1.2e+154) {
		tmp = (pow(k, m) / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = pow(k, (-1.0 + m)) * (a / k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -5e+156)
		tmp = Float64(a * (k ^ m));
	elseif (k <= 1.2e+154)
		tmp = Float64(Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)) * a);
	else
		tmp = Float64((k ^ Float64(-1.0 + m)) * Float64(a / k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -5e+156], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e+154], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, N[(-1.0 + m), $MachinePrecision]], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.99999999999999992e156

   1. Initial program 60.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.9

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -4.99999999999999992e156 < k < 1.20000000000000007e154

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 1.20000000000000007e154 < k

   1. Initial program 77.6%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot a}}{{k}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot \frac{a}{{k}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} \]

      6. exp-prod N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} \]

      8. log-rec N/A

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

      9. remove-double-neg N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} \]

      10. rem-exp-log N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 97.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{+156}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 28.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ (* 10.0 k) 1.0) (* k k))) 0.0)
   (* (* (* 99.0 k) a) k)
   (* (fma (fma 99.0 k -10.0) k 1.0) a)))

C

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 17.4%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 16.6%

       \[\leadsto \left(\left(99 \cdot k\right) \cdot a\right) \cdot k \]

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

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 38.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 34.2%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 6.4%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 48.9%

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

4. Final simplification 25.3%

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

Alternative 3: 17.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 8.7%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]
   12. Step-by-step derivation

   13. Applied rewrites 8.7%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]

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

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 38.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 34.2%

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

   10. Applied rewrites 35.6%

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

4. Final simplification 16.0%

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

Alternative 4: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7e-9) (* a (pow k m)) (* (pow k (+ -1.0 m)) (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-9) {
		tmp = a * pow(k, m);
	} else {
		tmp = pow(k, (-1.0 + m)) * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 7d-9) then
        tmp = a * (k ** m)
    else
        tmp = (k ** ((-1.0d0) + m)) * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-9) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = Math.pow(k, (-1.0 + m)) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7e-9:
		tmp = a * math.pow(k, m)
	else:
		tmp = math.pow(k, (-1.0 + m)) * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7e-9)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64((k ^ Float64(-1.0 + m)) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7e-9)
		tmp = a * (k ^ m);
	else
		tmp = (k ^ (-1.0 + m)) * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7e-9], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[Power[k, N[(-1.0 + m), $MachinePrecision]], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999998e-9

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.7

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 6.9999999999999998e-9 < k

   1. Initial program 87.0%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot a}}{{k}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot \frac{a}{{k}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} \]

      6. exp-prod N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} \]

      8. log-rec N/A

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

      9. remove-double-neg N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} \]

      10. rem-exp-log N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 89.7

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

   5. Applied rewrites 89.7%

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

   7. Applied rewrites 94.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.00025:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -1.1e-12)
     t_0
     (if (<= m 0.00025) (/ a (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -1.1e-12) {
		tmp = t_0;
	} else if (m <= 0.00025) {
		tmp = a / fma((10.0 + k), k, 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 <= -1.1e-12)
		tmp = t_0;
	elseif (m <= 0.00025)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.1e-12], t$95$0, If[LessEqual[m, 0.00025], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.09999999999999996e-12 or 2.5000000000000001e-4 < m

   1. Initial program 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.09999999999999996e-12 < m < 2.5000000000000001e-4

   1. Initial program 91.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 0.00025:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-2 + m\right)} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7e-9) (* a (pow k m)) (* (pow k (+ -2.0 m)) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-9) {
		tmp = a * pow(k, m);
	} else {
		tmp = pow(k, (-2.0 + m)) * 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 (k <= 7d-9) then
        tmp = a * (k ** m)
    else
        tmp = (k ** ((-2.0d0) + m)) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-9) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = Math.pow(k, (-2.0 + m)) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7e-9:
		tmp = a * math.pow(k, m)
	else:
		tmp = math.pow(k, (-2.0 + m)) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7e-9)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64((k ^ Float64(-2.0 + m)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7e-9)
		tmp = a * (k ^ m);
	else
		tmp = (k ^ (-2.0 + m)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7e-9], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[Power[k, N[(-2.0 + m), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999998e-9

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.7

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 6.9999999999999998e-9 < k

   1. Initial program 87.0%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot a}}{{k}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot \frac{a}{{k}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} \]

      6. exp-prod N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} \]

      8. log-rec N/A

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

      9. remove-double-neg N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} \]

      10. rem-exp-log N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 89.7

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

   5. Applied rewrites 89.7%

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 91.1%

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

   11. Applied rewrites 91.1%

       \[\leadsto \color{blue}{{k}^{\left(-2 + m\right)} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-2 + m\right)} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{99 \cdot \frac{\frac{a}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2000.0)
   (/ (* 99.0 (/ (/ a k) k)) (* k k))
   (if (<= m 0.95) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* 99.0 k) a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2000.0) {
		tmp = (99.0 * ((a / k) / k)) / (k * k);
	} else if (m <= 0.95) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((99.0 * k) * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2000.0)
		tmp = Float64(Float64(99.0 * Float64(Float64(a / k) / k)) / Float64(k * k));
	elseif (m <= 0.95)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2000.0], N[(N[(99.0 * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 79.6%

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

   13. Applied rewrites 79.6%

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

   if -2e3 < m < 0.94999999999999996

   1. Initial program 92.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 90.4%

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

   if 0.94999999999999996 < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 47.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{99 \cdot \frac{\frac{a}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2000.0)
   (* (/ 1.0 (* k k)) a)
   (if (<= m 0.95) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* 99.0 k) a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2000.0) {
		tmp = (1.0 / (k * k)) * a;
	} else if (m <= 0.95) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((99.0 * k) * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2000.0)
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	elseif (m <= 0.95)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2000.0], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot a}}{{k}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} \cdot \frac{a}{{k}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a}{{k}^{2}} \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} \]

      6. exp-prod N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} \]

      8. log-rec N/A

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

      9. remove-double-neg N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} \]

      10. rem-exp-log N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 76.3%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in m around 0

       \[\leadsto a \cdot \frac{1}{\color{blue}{{k}^{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 71.3%

       \[\leadsto a \cdot \frac{1}{\color{blue}{k \cdot k}} \]

   if -2e3 < m < 0.94999999999999996

   1. Initial program 92.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 90.4%

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

   if 0.94999999999999996 < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 47.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2000:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.95) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* 99.0 k) a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.95) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((99.0 * k) * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.95)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.95], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.94999999999999996

   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 m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   if 0.94999999999999996 < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 47.0%

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

Alternative 10: 44.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.68:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.68) (/ a (fma 10.0 k 1.0)) (* (* (* 99.0 k) a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.68) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((99.0 * k) * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.68)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.68], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.680000000000000049

   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 m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 40.1%

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

   if 0.680000000000000049 < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 47.0%

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

Alternative 11: 35.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.19:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.19) (fma (* -10.0 a) k a) (* (* (* 99.0 k) a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.19) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = ((99.0 * k) * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.19)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = Float64(Float64(Float64(99.0 * k) * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.19], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.19

   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 m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   10. Applied rewrites 25.7%

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

   if 0.19 < 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. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 47.0%

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

Alternative 12: 25.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1e-5) (fma (* -10.0 a) k a) (* (* -10.0 k) a)))

C

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

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1e-5)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1e-5], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.09999999999999988e-5

   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 m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.9%

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

   10. Applied rewrites 25.9%

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

   if 2.09999999999999988e-5 < m

   1. Initial program 82.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 7.0%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 21.9%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]
   12. Step-by-step derivation

   13. Applied rewrites 23.1%

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

Alternative 13: 8.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(-10 \cdot k\right) \cdot a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* (* -10.0 k) a))

C

double code(double a, double k, double m) {
	return (-10.0 * k) * 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 = ((-10.0d0) * k) * a
end function

Java

public static double code(double a, double k, double m) {
	return (-10.0 * k) * a;
}

Python

def code(a, k, m):
	return (-10.0 * k) * a

Julia

function code(a, k, m)
	return Float64(Float64(-10.0 * k) * a)
end

MATLAB

function tmp = code(a, k, m)
	tmp = (-10.0 * k) * a;
end

Wolfram

code[a_, k_, m_] := N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. lft-mult-inverse N/A

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

   7. associate-*l* N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 20.3%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 8.1%

    \[\leadsto \left(a \cdot k\right) \cdot -10 \]
12. Step-by-step derivation

13. Applied rewrites 8.5%

    \[\leadsto \left(-10 \cdot k\right) \cdot a \]
14. Add Preprocessing

Alternative 14: 8.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(-10 \cdot a\right) \cdot k \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* (* -10.0 a) k))

C

double code(double a, double k, double m) {
	return (-10.0 * a) * 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 = ((-10.0d0) * a) * k
end function

Java

public static double code(double a, double k, double m) {
	return (-10.0 * a) * k;
}

Python

def code(a, k, m):
	return (-10.0 * a) * k

Julia

function code(a, k, m)
	return Float64(Float64(-10.0 * a) * k)
end

MATLAB

function tmp = code(a, k, m)
	tmp = (-10.0 * a) * k;
end

Wolfram

code[a_, k_, m_] := N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. lft-mult-inverse N/A

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

   7. associate-*l* N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 20.3%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 8.1%

    \[\leadsto \left(a \cdot k\right) \cdot -10 \]
12. Step-by-step derivation

13. Applied rewrites 8.1%

    \[\leadsto \left(-10 \cdot a\right) \cdot k \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))