Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 98.2%

Time: 12.2s

Alternatives: 14

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) INFINITY)
     (/ t_0 (fma (+ k 10.0) k 1.0))
     (* (* k k) (* k (* a (fma k 1e-6 -1e-5)))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= ((double) INFINITY)) {
		tmp = t_0 / fma((k + 10.0), k, 1.0);
	} else {
		tmp = (k * k) * (k * (a * fma(k, 1e-6, -1e-5)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= Inf)
		tmp = Float64(t_0 / fma(Float64(k + 10.0), k, 1.0));
	else
		tmp = Float64(Float64(k * k) * Float64(k * Float64(a * fma(k, 1e-6, -1e-5))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{k}^{2}} \cdot k\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right)\right) \]

       12. distribute-lft-out N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       14. lower-fma.f64 100.0

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right)\right) \]

   13. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) INFINITY)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (* (* k k) (* k (* a (fma k 1e-6 -1e-5))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= ((double) INFINITY)) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (k * k) * (k * (a * fma(k, 1e-6, -1e-5)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= Inf)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64(k * k) * Float64(k * Float64(a * fma(k, 1e-6, -1e-5))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 98.0

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 1.6%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{k}^{2}} \cdot k\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right)\right) \]

       12. distribute-lft-out N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       14. lower-fma.f64 100.0

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right)\right) \]

   13. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -4.6e-5)
     t_0
     (if (<= m 1.02e-44) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -4.6e-5) {
		tmp = t_0;
	} else if (m <= 1.02e-44) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -4.6e-5)
		tmp = t_0;
	elseif (m <= 1.02e-44)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.6e-5], t$95$0, If[LessEqual[m, 1.02e-44], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -4.6e-5 or 1.0199999999999999e-44 < m

   1. Initial program 89.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.6e-5 < m < 1.0199999999999999e-44

   1. Initial program 94.1%

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

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{100}{k} + -10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, -k \cdot \left(a \cdot \mathsf{fma}\left(-9.9 \cdot 10^{-9}, k, 10^{-7}\right)\right), a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ (fma (/ a k) (+ (/ 100.0 k) -10.0) a) (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (*
      k
      (*
       (* k k)
       (fma
        k
        (- (* k (* a (fma -9.9e-9 k 1e-7))))
        (* a (fma k 1e-6 -1e-5))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = fma((a / k), ((100.0 / k) + -10.0), a) / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * ((k * k) * fma(k, -(k * (a * fma(-9.9e-9, k, 1e-7))), (a * fma(k, 1e-6, -1e-5))));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(fma(Float64(a / k), Float64(Float64(100.0 / k) + -10.0), a) / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(Float64(k * k) * fma(k, Float64(-Float64(k * Float64(a * fma(-9.9e-9, k, 1e-7)))), Float64(a * fma(k, 1e-6, -1e-5)))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(N[(N[(a / k), $MachinePrecision] * N[(N[(100.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(N[(k * k), $MachinePrecision] * N[(k * (-N[(k * N[(a * N[(-9.9e-9 * k + 1e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 43.2

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in k around inf

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

       1. metadata-eval N/A

          \[\leadsto \frac{\left(a + \color{blue}{\left(\mathsf{neg}\left(-100\right)\right)} \cdot \frac{a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{\left(a - -100 \cdot \frac{a}{{k}^{2}}\right)} - 10 \cdot \frac{a}{k}}{{k}^{2}} \]

       3. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{a - \left(-100 \cdot \frac{a}{{k}^{2}} + 10 \cdot \frac{a}{k}\right)}}{{k}^{2}} \]

       4. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a - \left(-100 \cdot \frac{a}{{k}^{2}} + 10 \cdot \frac{a}{k}\right)}{{k}^{2}}} \]

   11. Applied rewrites 65.5%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 46.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(\frac{-1}{100000000} \cdot a + \frac{1}{10000000000} \cdot a\right)\right) - \frac{1}{10000000} \cdot a\right) - \frac{-1}{1000000} \cdot a\right)\right)} \]

   13. Applied rewrites 76.0%

       \[\leadsto \color{blue}{k \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, -\left(a \cdot \mathsf{fma}\left(-9.9 \cdot 10^{-9}, k, 10^{-7}\right)\right) \cdot k, a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{100}{k} + -10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, -k \cdot \left(a \cdot \mathsf{fma}\left(-9.9 \cdot 10^{-9}, k, 10^{-7}\right)\right), a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{100}{k} + -10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ (fma (/ a k) (+ (/ 100.0 k) -10.0) a) (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (* (* k k) (* k (* a (fma k 1e-6 -1e-5)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = fma((a / k), ((100.0 / k) + -10.0), a) / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (k * k) * (k * (a * fma(k, 1e-6, -1e-5)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(fma(Float64(a / k), Float64(Float64(100.0 / k) + -10.0), a) / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(k * k) * Float64(k * Float64(a * fma(k, 1e-6, -1e-5))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(N[(N[(a / k), $MachinePrecision] * N[(N[(100.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 43.2

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in k around inf

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

       1. metadata-eval N/A

          \[\leadsto \frac{\left(a + \color{blue}{\left(\mathsf{neg}\left(-100\right)\right)} \cdot \frac{a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{\left(a - -100 \cdot \frac{a}{{k}^{2}}\right)} - 10 \cdot \frac{a}{k}}{{k}^{2}} \]

       3. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{a - \left(-100 \cdot \frac{a}{{k}^{2}} + 10 \cdot \frac{a}{k}\right)}}{{k}^{2}} \]

       4. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a - \left(-100 \cdot \frac{a}{{k}^{2}} + 10 \cdot \frac{a}{k}\right)}{{k}^{2}}} \]

   11. Applied rewrites 65.5%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 46.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{k}^{2}} \cdot k\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right)\right) \]

       12. distribute-lft-out N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       14. lower-fma.f64 74.1

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right)\right) \]

   13. Applied rewrites 74.1%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{100}{k} + -10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ a (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (* (* k k) (* k (* a (fma k 1e-6 -1e-5)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (k * k) * (k * (a * fma(k, 1e-6, -1e-5)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(k * k) * Float64(k * Float64(a * fma(k, 1e-6, -1e-5))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(k * N[(a * N[(k * 1e-6 + -1e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 46.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{3} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)} \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{k}^{2}} \cdot k\right) \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot \left(\frac{-1}{100000} \cdot a + \frac{1}{1000000} \cdot \left(a \cdot k\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(\frac{1}{1000000} \cdot \left(a \cdot k\right) + \frac{-1}{100000} \cdot a\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{\left(a \cdot k\right) \cdot \frac{1}{1000000}} + \frac{-1}{100000} \cdot a\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(\color{blue}{a \cdot \left(k \cdot \frac{1}{1000000}\right)} + \frac{-1}{100000} \cdot a\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \left(k \cdot \frac{1}{1000000}\right) + \color{blue}{a \cdot \frac{-1}{100000}}\right)\right) \]

       12. distribute-lft-out N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(k \cdot \frac{1}{1000000} + \frac{-1}{100000}\right)\right)}\right) \]

       14. lower-fma.f64 74.1

           \[\leadsto \left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)}\right)\right) \]

   13. Applied rewrites 74.1%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \left(k \cdot \left(a \cdot \mathsf{fma}\left(k, 10^{-6}, -1 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot 10^{-5} \cdot \left(a \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e+24)
   (/ a (* k k))
   (if (<= m 1.65)
     (/ a (fma k (+ k 10.0) 1.0))
     (* -1e-5 (* a (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e+24) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = -1e-5 * (a * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e+24)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(-1e-5 * Float64(a * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e+24], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-1e-5 * N[(a * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if -6.8000000000000001e24 < m < 1.6499999999999999

   1. Initial program 94.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 46.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{100000} \cdot \color{blue}{\left({k}^{3} \cdot a\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{100000} \cdot \color{blue}{\left({k}^{3} \cdot a\right)} \]

       4. cube-mult N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot a\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\color{blue}{\left(k \cdot {k}^{2}\right)} \cdot a\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot a\right) \]

       8. lower-*.f64 66.0

          \[\leadsto -1 \cdot 10^{-5} \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot a\right) \]

   13. Applied rewrites 66.0%

       \[\leadsto \color{blue}{-1 \cdot 10^{-5} \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot 10^{-5} \cdot \left(a \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot 10^{-5} \cdot \left(a \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.65e-30)
   (/ a (* k k))
   (if (<= m 1.65) (/ a (fma k 10.0 1.0)) (* -1e-5 (* a (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.65e-30) {
		tmp = a / (k * k);
	} else if (m <= 1.65) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = -1e-5 * (a * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.65e-30)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(-1e-5 * Float64(a * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.65e-30], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(-1e-5 * N[(a * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.64999999999999987e-30

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if -2.64999999999999987e-30 < m < 1.6499999999999999

   1. Initial program 95.1%

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

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   8. Applied rewrites 60.4%

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

   if 1.6499999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. flip-+ N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}, 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} + \color{blue}{-1}, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + -1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k}}, 1\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, -1 + \color{blue}{\frac{-1 \cdot \left(10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)\right)}{k}}, 1\right)} \]

   10. Applied rewrites 46.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(\frac{100 - k \cdot k}{1}, \color{blue}{-1 + \frac{-10 + \left(\frac{-1000}{k \cdot k} + \frac{-100}{k}\right)}{k}}, 1\right)} \]

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{100000} \cdot \left(a \cdot {k}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{100000} \cdot \color{blue}{\left({k}^{3} \cdot a\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{-1}{100000} \cdot \color{blue}{\left({k}^{3} \cdot a\right)} \]

       4. cube-mult N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot a\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\color{blue}{\left(k \cdot {k}^{2}\right)} \cdot a\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{-1}{100000} \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot a\right) \]

       8. lower-*.f64 66.0

          \[\leadsto -1 \cdot 10^{-5} \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot a\right) \]

   13. Applied rewrites 66.0%

       \[\leadsto \color{blue}{-1 \cdot 10^{-5} \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot 10^{-5} \cdot \left(a \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.65e-30)
   (/ a (* k k))
   (if (<= m 1.9) (/ a (fma k 10.0 1.0)) (* k (* a (fma 99.0 k -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.65e-30) {
		tmp = a / (k * k);
	} else if (m <= 1.9) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = k * (a * fma(99.0, k, -10.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.65e-30)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.9)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(k * Float64(a * fma(99.0, k, -10.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.65e-30], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.64999999999999987e-30

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if -2.64999999999999987e-30 < m < 1.8999999999999999

   1. Initial program 95.1%

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

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   8. Applied rewrites 60.4%

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

   if 1.8999999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({k}^{3} \cdot \left(-1 \cdot \frac{-10 \cdot \frac{a}{k} + 99 \cdot a}{k} + 980 \cdot a\right)\right)} \]

   11. Applied rewrites 36.4%

       \[\leadsto \color{blue}{-\left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(a, 980, \frac{a \cdot \left(\frac{10}{k} + -99\right)}{k}\right)} \]

   12. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 36.3

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

   14. Applied rewrites 36.3%

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

Alternative 10: 37.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0023:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.49:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.0023)
   (/ a (* k 10.0))
   (if (<= m 0.49) a (* k (* a (fma 99.0 k -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0023) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.49) {
		tmp = a;
	} else {
		tmp = k * (a * fma(99.0, k, -10.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.0023)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 0.49)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * fma(99.0, k, -10.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.0023], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.49], a, N[(k * N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0023

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 44.1

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

   8. Applied rewrites 44.1%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{10 \cdot k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 26.0

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

   11. Applied rewrites 26.0%

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

   if -0.0023 < m < 0.48999999999999999

   1. Initial program 94.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.6%

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 45.6

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 45.6%

       \[\leadsto \color{blue}{a} \]

   if 0.48999999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({k}^{3} \cdot \left(-1 \cdot \frac{-10 \cdot \frac{a}{k} + 99 \cdot a}{k} + 980 \cdot a\right)\right)} \]

   11. Applied rewrites 36.4%

       \[\leadsto \color{blue}{-\left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(a, 980, \frac{a \cdot \left(\frac{10}{k} + -99\right)}{k}\right)} \]

   12. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. lower-*.f64 N/A

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

       8. lower-fma.f64 36.3

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

   14. Applied rewrites 36.3%

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

Alternative 11: 48.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.9) (/ a (* k k)) (* k (* a (fma 99.0 k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.9) {
		tmp = a / (k * k);
	} else {
		tmp = k * (a * fma(99.0, k, -10.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.9)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(k * Float64(a * fma(99.0, k, -10.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.9], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.8999999999999999

   1. Initial program 97.1%

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

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 54.9

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

   8. Applied rewrites 54.9%

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

   if 1.8999999999999999 < m

   1. Initial program 80.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), a \cdot 99\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({k}^{3} \cdot \left(-1 \cdot \frac{-10 \cdot \frac{a}{k} + 99 \cdot a}{k} + 980 \cdot a\right)\right)} \]

   11. Applied rewrites 36.4%

       \[\leadsto \color{blue}{-\left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(a, 980, \frac{a \cdot \left(\frac{10}{k} + -99\right)}{k}\right)} \]

   12. Taylor expanded in k around 0

       \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a\right)} \]

       3. metadata-eval N/A

          \[\leadsto k \cdot \left(99 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a\right) \]

       4. *-commutative N/A

          \[\leadsto k \cdot \left(99 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a\right) \]

       5. associate-*r* N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(99 \cdot k\right) \cdot a} + -10 \cdot a\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto k \cdot \color{blue}{\left(a \cdot \left(99 \cdot k + -10\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto k \cdot \color{blue}{\left(a \cdot \left(99 \cdot k + -10\right)\right)} \]

       8. lower-fma.f64 36.3

          \[\leadsto k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(99, k, -10\right)}\right) \]

   14. Applied rewrites 36.3%

       \[\leadsto \color{blue}{k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 31.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.49:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.49) a (* k (* a (fma 99.0 k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.49) {
		tmp = a;
	} else {
		tmp = k * (a * fma(99.0, k, -10.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.49)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * fma(99.0, k, -10.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.49], a, N[(k * N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.48999999999999999

   1. Initial program 97.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 63.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 24.6

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 24.6%

       \[\leadsto \color{blue}{a} \]

   if 0.48999999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), a \cdot 99\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({k}^{3} \cdot \left(-1 \cdot \frac{-10 \cdot \frac{a}{k} + 99 \cdot a}{k} + 980 \cdot a\right)\right)} \]

   11. Applied rewrites 36.4%

       \[\leadsto \color{blue}{-\left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(a, 980, \frac{a \cdot \left(\frac{10}{k} + -99\right)}{k}\right)} \]

   12. Taylor expanded in k around 0

       \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a\right)} \]

       3. metadata-eval N/A

          \[\leadsto k \cdot \left(99 \cdot \left(a \cdot k\right) + \color{blue}{-10} \cdot a\right) \]

       4. *-commutative N/A

          \[\leadsto k \cdot \left(99 \cdot \color{blue}{\left(k \cdot a\right)} + -10 \cdot a\right) \]

       5. associate-*r* N/A

          \[\leadsto k \cdot \left(\color{blue}{\left(99 \cdot k\right) \cdot a} + -10 \cdot a\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto k \cdot \color{blue}{\left(a \cdot \left(99 \cdot k + -10\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto k \cdot \color{blue}{\left(a \cdot \left(99 \cdot k + -10\right)\right)} \]

       8. lower-fma.f64 36.3

          \[\leadsto k \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(99, k, -10\right)}\right) \]

   14. Applied rewrites 36.3%

       \[\leadsto \color{blue}{k \cdot \left(a \cdot \mathsf{fma}\left(99, k, -10\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 24.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.49:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.49) a (* k (* a -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.49) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.49d0) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.49) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.49:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.49)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.49)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.49], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.48999999999999999

   1. Initial program 97.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 63.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto \frac{a}{\color{blue}{1}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 24.6

         \[\leadsto \color{blue}{a} \]

   10. Applied rewrites 24.6%

       \[\leadsto \color{blue}{a} \]

   if 0.48999999999999999 < m

   1. Initial program 80.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(k, 10 \cdot \left(-98 \cdot a\right), a \cdot 99\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({k}^{3} \cdot \left(-1 \cdot \frac{-10 \cdot \frac{a}{k} + 99 \cdot a}{k} + 980 \cdot a\right)\right)} \]

   11. Applied rewrites 36.4%

       \[\leadsto \color{blue}{-\left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(a, 980, \frac{a \cdot \left(\frac{10}{k} + -99\right)}{k}\right)} \]

   12. Taylor expanded in k around 0

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-10 \cdot a\right) \cdot k} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(-10 \cdot a\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(-10 \cdot a\right)} \]

       4. *-commutative N/A

          \[\leadsto k \cdot \color{blue}{\left(a \cdot -10\right)} \]

       5. lower-*.f64 21.9

          \[\leadsto k \cdot \color{blue}{\left(a \cdot -10\right)} \]

   14. Applied rewrites 21.9%

       \[\leadsto \color{blue}{k \cdot \left(a \cdot -10\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 20.0% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.1%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 41.6

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Applied rewrites 41.6%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{a}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 17.1%

   \[\leadsto \frac{a}{\color{blue}{1}} \]
9. Step-by-step derivation

   1. /-rgt-identity 17.1

      \[\leadsto \color{blue}{a} \]

10. Applied rewrites 17.1%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))