Falkner and Boettcher, Appendix A

Percentage Accurate: 91.0% → 99.7%

Time: 12.5s

Alternatives: 16

Speedup: 1.1×

• 22.0% 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: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 6.4 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k}{t\_0} + \frac{10}{t\_0}, \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 6.4e-27)
     t_0
     (/ 1.0 (fma k (+ (/ k t_0) (/ 10.0 t_0)) (/ 1.0 t_0))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 6.4e-27) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(k, ((k / t_0) + (10.0 / t_0)), (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 6.4e-27)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(k, Float64(Float64(k / t_0) + Float64(10.0 / t_0)), Float64(1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.4e-27], t$95$0, N[(1.0 / N[(k * N[(N[(k / t$95$0), $MachinePrecision] + N[(10.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.39999999999999982e-27

   1. Initial program 95.9%

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

   3. Taylor expanded in 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 6.39999999999999982e-27 < k

   1. Initial program 82.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 82.4

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 2: 38.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (* k k) (+ 1.0 (* k 10.0))))))
   (if (<= t_0 4e-283)
     (/ a (* k (+ k 10.0)))
     (if (<= t_0 4e+287) (/ a (fma k 10.0 1.0)) (/ a (* k k))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)));
	double tmp;
	if (t_0 <= 4e-283) {
		tmp = a / (k * (k + 10.0));
	} else if (t_0 <= 4e+287) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))))
	tmp = 0.0
	if (t_0 <= 4e-283)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (t_0 <= 4e+287)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-283], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+287], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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))) < 3.99999999999999979e-283

   1. Initial program 96.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 40.0%

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

   if 3.99999999999999979e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.0000000000000003e287

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 71.5%

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

   if 4.0000000000000003e287 < (/.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 59.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 22.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ 1.0 (* k 10.0))))))
   (if (<= t_1 4e-283) t_0 (if (<= t_1 4e+287) (/ a (fma k 10.0 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)));
	double tmp;
	if (t_1 <= 4e-283) {
		tmp = t_0;
	} else if (t_1 <= 4e+287) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))))
	tmp = 0.0
	if (t_1 <= 4e-283)
		tmp = t_0;
	elseif (t_1 <= 4e+287)
		tmp = Float64(a / fma(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[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-283], t$95$0, If[LessEqual[t$95$1, 4e+287], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999979e-283 or 4.0000000000000003e287 < (/.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 88.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 39.7%

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

   if 3.99999999999999979e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.0000000000000003e287

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 71.5%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+287}:\\ \;\;\;\;a \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ 1.0 (* k 10.0))))))
   (if (<= t_1 4e-283) t_0 (if (<= t_1 4e+287) (* a 1.0) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)));
	double tmp;
	if (t_1 <= 4e-283) {
		tmp = t_0;
	} else if (t_1 <= 4e+287) {
		tmp = a * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (k * k)
    t_1 = (a * (k ** m)) / ((k * k) + (1.0d0 + (k * 10.0d0)))
    if (t_1 <= 4d-283) then
        tmp = t_0
    else if (t_1 <= 4d+287) then
        tmp = a * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * Math.pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)));
	double tmp;
	if (t_1 <= 4e-283) {
		tmp = t_0;
	} else if (t_1 <= 4e+287) {
		tmp = a * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	t_1 = (a * math.pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)))
	tmp = 0
	if t_1 <= 4e-283:
		tmp = t_0
	elif t_1 <= 4e+287:
		tmp = a * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))))
	tmp = 0.0
	if (t_1 <= 4e-283)
		tmp = t_0;
	elseif (t_1 <= 4e+287)
		tmp = Float64(a * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	t_1 = (a * (k ^ m)) / ((k * k) + (1.0 + (k * 10.0)));
	tmp = 0.0;
	if (t_1 <= 4e-283)
		tmp = t_0;
	elseif (t_1 <= 4e+287)
		tmp = a * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-283], t$95$0, If[LessEqual[t$95$1, 4e+287], N[(a * 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999979e-283 or 4.0000000000000003e287 < (/.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 88.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 39.7%

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

   if 3.99999999999999979e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.0000000000000003e287

   1. Initial program 99.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 70.7%

      \[\leadsto a \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 4 \cdot 10^{+287}:\\ \;\;\;\;a \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 22.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (* k k) (+ 1.0 (* k 10.0)))) 1e-306)
   (/ a (* k 10.0))
   (fma a (* k -10.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)))) <= 1e-306) {
		tmp = a / (k * 10.0);
	} else {
		tmp = fma(a, (k * -10.0), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0)))) <= 1e-306)
		tmp = Float64(a / Float64(k * 10.0));
	else
		tmp = fma(a, Float64(k * -10.0), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-306], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000003e-306

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 18.1%

       \[\leadsto \frac{a}{k \cdot 10} \]

   if 1.00000000000000003e-306 < (/.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 75.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 39.5

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

   5. Applied rewrites 39.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 32.9%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.5e-43)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (if (<= m 0.04)
     (/ 1.0 (fma k (+ (/ 10.0 a) (/ k a)) (/ 1.0 a)))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.5e-43) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else if (m <= 0.04) {
		tmp = 1.0 / fma(k, ((10.0 / a) + (k / a)), (1.0 / a));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.5e-43)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	elseif (m <= 0.04)
		tmp = Float64(1.0 / fma(k, Float64(Float64(10.0 / a) + Float64(k / a)), Float64(1.0 / a)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.5e-43], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.04], N[(1.0 / N[(k * N[(N[(10.0 / a), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.50000000000000068e-43

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -7.50000000000000068e-43 < m < 0.0400000000000000008

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.9

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 98.7%

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

   if 0.0400000000000000008 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.5e-5)
   (* a (/ (pow k m) (* k k)))
   (if (<= m 0.04)
     (/ 1.0 (fma k (+ (/ 10.0 a) (/ k a)) (/ 1.0 a)))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.5e-5) {
		tmp = a * (pow(k, m) / (k * k));
	} else if (m <= 0.04) {
		tmp = 1.0 / fma(k, ((10.0 / a) + (k / a)), (1.0 / a));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.5e-5)
		tmp = Float64(a * Float64((k ^ m) / Float64(k * k)));
	elseif (m <= 0.04)
		tmp = Float64(1.0 / fma(k, Float64(Float64(10.0 / a) + Float64(k / a)), Float64(1.0 / a)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.5e-5], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.04], N[(1.0 / N[(k * N[(N[(10.0 / a), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.50000000000000028e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -4.50000000000000028e-5 < m < 0.0400000000000000008

   1. Initial program 93.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.3

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.8%

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

   if 0.0400000000000000008 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -51000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -51000.0)
     t_0
     (if (<= m 0.04) (/ 1.0 (fma k (+ (/ 10.0 a) (/ k a)) (/ 1.0 a))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -51000.0) {
		tmp = t_0;
	} else if (m <= 0.04) {
		tmp = 1.0 / fma(k, ((10.0 / a) + (k / a)), (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -51000.0)
		tmp = t_0;
	elseif (m <= 0.04)
		tmp = Float64(1.0 / fma(k, Float64(Float64(10.0 / a) + Float64(k / a)), Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -51000.0], t$95$0, If[LessEqual[m, 0.04], N[(1.0 / N[(k * N[(N[(10.0 / a), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -51000 or 0.0400000000000000008 < m

   1. Initial program 87.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 -51000 < m < 0.0400000000000000008

   1. Initial program 93.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.5

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.1%

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

Alternative 9: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a \cdot 99}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{10}{a} + \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ (* a 99.0) (* k (* k (* k k))))
   (if (<= m 1.06e+24)
     (/ 1.0 (fma k (+ (/ 10.0 a) (/ k a)) (/ 1.0 a)))
     (fma k (fma a -10.0 (* k (* a 99.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = (a * 99.0) / (k * (k * (k * k)));
	} else if (m <= 1.06e+24) {
		tmp = 1.0 / fma(k, ((10.0 / a) + (k / a)), (1.0 / a));
	} else {
		tmp = fma(k, fma(a, -10.0, (k * (a * 99.0))), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(Float64(a * 99.0) / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 1.06e+24)
		tmp = Float64(1.0 / fma(k, Float64(Float64(10.0 / a) + Float64(k / a)), Float64(1.0 / a)));
	else
		tmp = fma(k, fma(a, -10.0, Float64(k * Float64(a * 99.0))), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(N[(a * 99.0), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.06e+24], N[(1.0 / N[(k * N[(N[(10.0 / a), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0 + N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 66.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto 99 \cdot \frac{a}{\color{blue}{{k}^{4}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.3%

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

   if -51000 < m < 1.06e24

   1. Initial program 93.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.5

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 96.2%

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

   if 1.06e24 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.4%

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

Alternative 10: 66.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a \cdot 99}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ (* a 99.0) (* k (* k (* k k))))
   (if (<= m 2.0)
     (* a (/ 1.0 (fma k k (fma k 10.0 1.0))))
     (fma k (fma a -10.0 (* k (* a 99.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = (a * 99.0) / (k * (k * (k * k)));
	} else if (m <= 2.0) {
		tmp = a * (1.0 / fma(k, k, fma(k, 10.0, 1.0)));
	} else {
		tmp = fma(k, fma(a, -10.0, (k * (a * 99.0))), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(Float64(a * 99.0) / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 2.0)
		tmp = Float64(a * Float64(1.0 / fma(k, k, fma(k, 10.0, 1.0))));
	else
		tmp = fma(k, fma(a, -10.0, Float64(k * Float64(a * 99.0))), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(N[(a * 99.0), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a * N[(1.0 / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0 + N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 66.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto 99 \cdot \frac{a}{\color{blue}{{k}^{4}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.3%

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

   if -51000 < m < 2

   1. Initial program 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.6

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 91.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 91.6

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

   9. Applied rewrites 91.6%

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

   if 2 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a \cdot 99}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ a (* k k))
   (if (<= m 2.0)
     (* a (/ 1.0 (fma k k (fma k 10.0 1.0))))
     (fma k (fma a -10.0 (* k (* a 99.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = a / (k * k);
	} else if (m <= 2.0) {
		tmp = a * (1.0 / fma(k, k, fma(k, 10.0, 1.0)));
	} else {
		tmp = fma(k, fma(a, -10.0, (k * (a * 99.0))), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.0)
		tmp = Float64(a * Float64(1.0 / fma(k, k, fma(k, 10.0, 1.0))));
	else
		tmp = fma(k, fma(a, -10.0, Float64(k * Float64(a * 99.0))), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a * N[(1.0 / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0 + N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.4%

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

   if -51000 < m < 2

   1. Initial program 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.6

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 91.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 91.6

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

   9. Applied rewrites 91.6%

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

   if 2 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ a (* k k))
   (if (<= m 2.0)
     (* a (/ 1.0 (fma k (+ k 10.0) 1.0)))
     (fma k (fma a -10.0 (* k (* a 99.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = a / (k * k);
	} else if (m <= 2.0) {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = fma(k, fma(a, -10.0, (k * (a * 99.0))), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.0)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = fma(k, fma(a, -10.0, Float64(k * Float64(a * 99.0))), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0 + N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.4%

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

   if -51000 < m < 2

   1. Initial program 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.6

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 91.6%

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

   if 2 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ a (* k k))
   (if (<= m 2.0)
     (/ a (fma k (+ k 10.0) 1.0))
     (fma k (fma a -10.0 (* k (* a 99.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = a / (k * k);
	} else if (m <= 2.0) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = fma(k, fma(a, -10.0, (k * (a * 99.0))), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.0)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = fma(k, fma(a, -10.0, Float64(k * Float64(a * 99.0))), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0 + N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.4%

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

   if -51000 < m < 2

   1. Initial program 93.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 2 < m

   1. Initial program 76.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, k \cdot \left(a \cdot 99\right)\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -51000.0)
   (/ a (* k k))
   (if (<= m 2e+24) (/ a (fma k (+ k 10.0) 1.0)) (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -51000.0) {
		tmp = a / (k * k);
	} else if (m <= 2e+24) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -51000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2e+24)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -51000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2e+24], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -51000

   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 33.3

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.4%

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

   if -51000 < m < 2e24

   1. Initial program 92.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 2e24 < m

   1. Initial program 77.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 20.1%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -51000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{+24}:\\ \;\;\;\;a \cdot 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2e+24) (* a 1.0) (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2e+24) {
		tmp = a * 1.0;
	} else {
		tmp = a * (k * -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 <= 2d+24) then
        tmp = a * 1.0d0
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2e+24) {
		tmp = a * 1.0;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2e+24:
		tmp = a * 1.0
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2e+24)
		tmp = Float64(a * 1.0);
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2e+24)
		tmp = a * 1.0;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2e+24], N[(a * 1.0), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2e24

   1. Initial program 95.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 27.2%

      \[\leadsto a \cdot 1 \]

   if 2e24 < m

   1. Initial program 77.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 8.9%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 20.1%

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

Alternative 16: 19.9% accurate, 22.3× speedup

Math

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

FPCore

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

C

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

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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in m around 0

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

8. Applied rewrites 19.9%

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

Reproduce

herbie shell --seed 2024222 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))