Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 99.7%

Time: 11.7s

Alternatives: 14

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(k, \frac{k}{t\_0} + \frac{10}{t\_0}, \frac{1}{t\_0}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 5e-90)
     t_0
     (pow (fma k (+ (/ k t_0) (/ 10.0 t_0)) (/ 1.0 t_0)) -1.0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 5e-90) {
		tmp = t_0;
	} else {
		tmp = pow(fma(k, ((k / t_0) + (10.0 / t_0)), (1.0 / t_0)), -1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 5e-90)
		tmp = t_0;
	else
		tmp = fma(k, Float64(Float64(k / t_0) + Float64(10.0 / t_0)), Float64(1.0 / t_0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5e-90], t$95$0, N[Power[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], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000019e-90

   1. Initial program 97.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.00000000000000019e-90 < k

   1. Initial program 79.4%

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

      1. lift-/.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. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 79.4

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 2: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+189)
     (/ 1.0 (fma k (* (/ (pow k (- m)) a) (+ k 10.0)) (/ 1.0 a)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+189) {
		tmp = 1.0 / fma(k, ((pow(k, -m) / a) * (k + 10.0)), (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+189)
		tmp = Float64(1.0 / fma(k, Float64(Float64((k ^ Float64(-m)) / a) * Float64(k + 10.0)), Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+189], N[(1.0 / N[(k * N[(N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2e189

   1. Initial program 94.8%

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

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 94.8

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 77.6

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

   7. Applied rewrites 77.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 77.6

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 90.6%

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

   if 2e189 < (/.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 70.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{\left(-m\right)}}{a}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-81}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, t\_0 \cdot \left(k + 10\right), t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (pow k (- m)) a)))
   (if (<= k 2e-81) (* a (pow k m)) (/ 1.0 (fma k (* t_0 (+ k 10.0)) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m) / a;
	double tmp;
	if (k <= 2e-81) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / fma(k, (t_0 * (k + 10.0)), t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ Float64(-m)) / a)
	tmp = 0.0
	if (k <= 2e-81)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / fma(k, Float64(t_0 * Float64(k + 10.0)), t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[k, 2e-81], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(t$95$0 * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e-81

   1. Initial program 97.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.9999999999999999e-81 < k

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 79.2

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.9

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

   7. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -9e+14)
     t_0
     (if (<= m 2.25e-7) (/ 1.0 (fma k (/ (+ k 10.0) 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 <= -9e+14) {
		tmp = t_0;
	} else if (m <= 2.25e-7) {
		tmp = 1.0 / fma(k, ((k + 10.0) / 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 <= -9e+14)
		tmp = t_0;
	elseif (m <= 2.25e-7)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / 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, -9e+14], t$95$0, If[LessEqual[m, 2.25e-7], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -9e14 or 2.2499999999999999e-7 < m

   1. Initial program 90.2%

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

   3. Taylor expanded in k around 0

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

      1. 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 -9e14 < m < 2.2499999999999999e-7

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 99.1%

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

Alternative 5: 67.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(t\_0, \left(k + 10\right) \cdot \left(k \cdot t\_0\right), 1\right)} \cdot 1\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -1.8e+17)
     (* (/ a (fma t_0 (* (+ k 10.0) (* k t_0)) 1.0)) 1.0)
     (if (<= m 57000000000.0)
       (/ 1.0 (fma k (/ (+ k 10.0) a) (/ 1.0 a)))
       (* a (* -980.0 (* k (* k k))))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -1.8e+17) {
		tmp = (a / fma(t_0, ((k + 10.0) * (k * t_0)), 1.0)) * 1.0;
	} else if (m <= 57000000000.0) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a), (1.0 / a));
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -1.8e+17)
		tmp = Float64(Float64(a / fma(t_0, Float64(Float64(k + 10.0) * Float64(k * t_0)), 1.0)) * 1.0);
	elseif (m <= 57000000000.0)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a), Float64(1.0 / a)));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.8e+17], N[(N[(a / N[(t$95$0 * N[(N[(k + 10.0), $MachinePrecision] * N[(k * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8e17

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 66.7%

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

   if -1.8e17 < m < 5.7e10

   1. Initial program 88.3%

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

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.2

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 98.7%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 94.7%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k \cdot \left(k + 10\right), \left(k + 10\right) \cdot \left(k \cdot \left(k \cdot \left(k + 10\right)\right)\right), 1\right)} \cdot 1\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a - \frac{-\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (/ (- a (/ (- (/ (* a 99.0) k)) k)) (* k k))
   (if (<= m 57000000000.0)
     (/ 1.0 (fma k (/ (+ k 10.0) a) (/ 1.0 a)))
     (* a (* -980.0 (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = (a - (-((a * 99.0) / k) / k)) / (k * k);
	} else if (m <= 57000000000.0) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a), (1.0 / a));
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(Float64(a - Float64(Float64(-Float64(Float64(a * 99.0) / k)) / k)) / Float64(k * k));
	elseif (m <= 57000000000.0)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a), Float64(1.0 / a)));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(N[(a - N[((-N[(N[(a * 99.0), $MachinePrecision] / k), $MachinePrecision]) / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 36.5

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

   5. Applied rewrites 36.5%

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a + -1 \cdot \frac{\left(-20 \cdot a + -1 \cdot \frac{99 \cdot a - \left(-30 \cdot \left(-20 \cdot a - -30 \cdot a\right) + 300 \cdot a\right)}{k}\right) - -30 \cdot a}{k}}{\color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.9%

       \[\leadsto \frac{a - \frac{\frac{a \cdot 99 - \mathsf{fma}\left(a, -300, a \cdot 300\right)}{-k} + 10 \cdot a}{k}}{\color{blue}{k \cdot k}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{a - \frac{-1 \cdot \frac{99 \cdot a - \left(-300 \cdot a + 300 \cdot a\right)}{k}}{k}}{k \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.5%

       \[\leadsto \frac{a - \frac{-\frac{a \cdot 99}{k}}{k}}{k \cdot k} \]

   if -9e14 < m < 5.7e10

   1. Initial program 88.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 95.7%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a - \frac{-\frac{a \cdot 99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* a (/ 1.0 (* k k)))
   (if (<= m 57000000000.0)
     (/ 1.0 (fma k (/ (+ k 10.0) a) (/ 1.0 a)))
     (* a (* -980.0 (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 57000000000.0) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a), (1.0 / a));
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 57000000000.0)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a), Float64(1.0 / a)));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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%

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

   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 36.5%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 5.7e10

   1. Initial program 88.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 88.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 95.7%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (* a (/ 1.0 (* k k)))
   (if (<= m 57000000000.0)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (* -980.0 (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 57000000000.0) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 57000000000.0)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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%

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

   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 36.5%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if -9e14 < m < 5.7e10

   1. Initial program 88.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9e+14)
   (/ a (* k k))
   (if (<= m 57000000000.0)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (* -980.0 (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+14) {
		tmp = a / (k * k);
	} else if (m <= 57000000000.0) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 57000000000.0)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9e14

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.5%

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

   if -9e14 < m < 5.7e10

   1. Initial program 88.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e-60)
   (/ a (* k k))
   (if (<= m 57000000000.0)
     (/ a (fma k 10.0 1.0))
     (* a (* -980.0 (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e-60) {
		tmp = a / (k * k);
	} else if (m <= 57000000000.0) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a * (-980.0 * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e-60)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 57000000000.0)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a * Float64(-980.0 * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e-60], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 57000000000.0], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-980.0 * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e-60

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 60.3%

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

   if -3.6e-60 < m < 5.7e10

   1. Initial program 87.3%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   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 65.1%

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

   if 5.7e10 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-10 \cdot a + k \cdot \left(99 \cdot a + k \cdot \left(20 \cdot a - 1000 \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.9%

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, a \cdot -980, a \cdot 99\right), a \cdot -10\right)}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto -980 \cdot \left(a \cdot \color{blue}{{k}^{3}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto a \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{-980}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 57000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-980 \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 990000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e-60)
   (/ a (* k k))
   (if (<= m 990000000000.0) (/ a (fma k 10.0 1.0)) (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e-60) {
		tmp = a / (k * k);
	} else if (m <= 990000000000.0) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e-60)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 990000000000.0)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e-60], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 990000000000.0], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e-60

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 60.3%

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

   if -3.6e-60 < m < 9.9e11

   1. Initial program 87.3%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   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 65.1%

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

   if 9.9e11 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   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 4.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 13.7%

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

Alternative 12: 46.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1.05 \cdot 10^{-285}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.082:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -1.05e-285) t_0 (if (<= k 0.082) (fma a (* k -10.0) a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -1.05e-285) {
		tmp = t_0;
	} else if (k <= 0.082) {
		tmp = fma(a, (k * -10.0), a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -1.05e-285)
		tmp = t_0;
	elseif (k <= 0.082)
		tmp = fma(a, Float64(k * -10.0), a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.05e-285], t$95$0, If[LessEqual[k, 0.082], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.04999999999999992e-285 or 0.0820000000000000034 < k

   1. Initial program 83.3%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 38.3%

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

   if -1.04999999999999992e-285 < k < 0.0820000000000000034

   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. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 54.3

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

   5. Applied rewrites 54.3%

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

   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 53.4%

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

Alternative 13: 25.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 990000000000:\\ \;\;\;\;a \cdot 1\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 990000000000.0) (* a 1.0) (* k (* a -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 990000000000.0) {
		tmp = a * 1.0;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 990000000000.0d0) then
        tmp = a * 1.0d0
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 990000000000.0) {
		tmp = a * 1.0;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 990000000000.0:
		tmp = a * 1.0
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 990000000000.0)
		tmp = Float64(a * 1.0);
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 990000000000.0)
		tmp = a * 1.0;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 990000000000.0], N[(a * 1.0), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.9e11

   1. Initial program 92.9%

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

   3. Taylor expanded in 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 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 34.0%

      \[\leadsto a \cdot 1 \]

   if 9.9e11 < m

   1. Initial program 84.2%

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

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

      15. associate-*l* N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   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 4.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 13.7%

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

Alternative 14: 19.7% 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 89.7%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 84.2

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

5. Applied rewrites 84.2%

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

6. Taylor expanded in m around 0

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

8. Applied rewrites 22.8%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))