Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 99.9%

Time: 10.0s

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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ t_1 := \frac{t\_0}{k \cdot k + \left(10 \cdot k + 1\right)}\\ t_2 := {\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{t\_2}{k}, -10, t\_2\right) \cdot a\_m}{k}}{k}\\ \mathbf{elif}\;t\_1 \leq 10^{+278}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m))
        (t_1 (/ t_0 (+ (* k k) (+ (* 10.0 k) 1.0))))
        (t_2 (pow (/ -1.0 (/ -1.0 k)) m)))
   (*
    a_s
    (if (<= t_1 0.0)
      (/ (/ (* (fma (/ t_2 k) -10.0 t_2) a_m) k) k)
      (if (<= t_1 1e+278) (/ t_0 (fma k 10.0 (fma k k 1.0))) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double t_1 = t_0 / ((k * k) + ((10.0 * k) + 1.0));
	double t_2 = pow((-1.0 / (-1.0 / k)), m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((fma((t_2 / k), -10.0, t_2) * a_m) / k) / k;
	} else if (t_1 <= 1e+278) {
		tmp = t_0 / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	t_1 = Float64(t_0 / Float64(Float64(k * k) + Float64(Float64(10.0 * k) + 1.0)))
	t_2 = Float64(-1.0 / Float64(-1.0 / k)) ^ m
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(fma(Float64(t_2 / k), -10.0, t_2) * a_m) / k) / k);
	elseif (t_1 <= 1e+278)
		tmp = Float64(t_0 / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(-1.0 / N[(-1.0 / k), $MachinePrecision]), $MachinePrecision], m], $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(t$95$2 / k), $MachinePrecision] * -10.0 + t$95$2), $MachinePrecision] * a$95$m), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$1, 1e+278], N[(t$95$0 / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.4%

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

   3. Taylor expanded in k around -inf

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

   5. Applied rewrites 72.1%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.99999999999999964e277

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 9.99999999999999964e277 < (/.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 51.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{{\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}}{k}, -10, {\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}\right) \cdot a}{k}}{k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{+278}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{{k}^{m} \cdot a\_m}{k \cdot k + \left(10 \cdot k + 1\right)}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;\frac{a\_m}{\left(10 + k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ (* (pow k m) a_m) (+ (* k k) (+ (* 10.0 k) 1.0)))))
   (*
    a_s
    (if (<= t_0 1e-315)
      (/ a_m (* (+ 10.0 k) k))
      (if (<= t_0 5e+295)
        (/ a_m (fma 10.0 k 1.0))
        (if (<= t_0 INFINITY)
          (/ a_m (* k k))
          (fma (* (fma 99.0 k -10.0) a_m) k a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = (pow(k, m) * a_m) / ((k * k) + ((10.0 * k) + 1.0));
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = a_m / ((10.0 + k) * k);
	} else if (t_0 <= 5e+295) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a_m / (k * k);
	} else {
		tmp = fma((fma(99.0, k, -10.0) * a_m), k, a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(Float64((k ^ m) * a_m) / Float64(Float64(k * k) + Float64(Float64(10.0 * k) + 1.0)))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(a_m / Float64(Float64(10.0 + k) * k));
	elseif (t_0 <= 5e+295)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a_m / Float64(k * k));
	else
		tmp = fma(Float64(fma(99.0, k, -10.0) * a_m), k, a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$0, 1e-315], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+295], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 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))) < 9.999999985e-316

   1. Initial program 97.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 31.8%

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

   if 9.999999985e-316 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.99999999999999991e295

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 68.0%

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

   if 4.99999999999999991e295 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 43.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 77.1%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{-315}:\\ \;\;\;\;\frac{a}{\left(10 + k\right) \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ t_1 := \frac{{k}^{m} \cdot a\_m}{k \cdot k + \left(10 \cdot k + 1\right)}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{-315}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k k)))
        (t_1 (/ (* (pow k m) a_m) (+ (* k k) (+ (* 10.0 k) 1.0)))))
   (*
    a_s
    (if (<= t_1 1e-315)
      t_0
      (if (<= t_1 5e+295)
        (/ a_m (fma 10.0 k 1.0))
        (if (<= t_1 INFINITY) t_0 (fma (* (fma 99.0 k -10.0) a_m) k a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double t_1 = (pow(k, m) * a_m) / ((k * k) + ((10.0 * k) + 1.0));
	double tmp;
	if (t_1 <= 1e-315) {
		tmp = t_0;
	} else if (t_1 <= 5e+295) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((fma(99.0, k, -10.0) * a_m), k, a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * k))
	t_1 = Float64(Float64((k ^ m) * a_m) / Float64(Float64(k * k) + Float64(Float64(10.0 * k) + 1.0)))
	tmp = 0.0
	if (t_1 <= 1e-315)
		tmp = t_0;
	elseif (t_1 <= 5e+295)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(fma(99.0, k, -10.0) * a_m), k, a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 1e-315], t$95$0, If[LessEqual[t$95$1, 5e+295], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.999999985e-316 or 4.99999999999999991e295 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 97.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 36.1%

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

   if 9.999999985e-316 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.99999999999999991e295

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 68.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 77.1%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{-315}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ t_1 := \frac{{k}^{m} \cdot a\_m}{k \cdot k + \left(10 \cdot k + 1\right)}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{-315}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;1 \cdot a\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k k)))
        (t_1 (/ (* (pow k m) a_m) (+ (* k k) (+ (* 10.0 k) 1.0)))))
   (*
    a_s
    (if (<= t_1 1e-315)
      t_0
      (if (<= t_1 5e+295)
        (* 1.0 a_m)
        (if (<= t_1 INFINITY) t_0 (fma (* (fma 99.0 k -10.0) a_m) k a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double t_1 = (pow(k, m) * a_m) / ((k * k) + ((10.0 * k) + 1.0));
	double tmp;
	if (t_1 <= 1e-315) {
		tmp = t_0;
	} else if (t_1 <= 5e+295) {
		tmp = 1.0 * a_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((fma(99.0, k, -10.0) * a_m), k, a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * k))
	t_1 = Float64(Float64((k ^ m) * a_m) / Float64(Float64(k * k) + Float64(Float64(10.0 * k) + 1.0)))
	tmp = 0.0
	if (t_1 <= 1e-315)
		tmp = t_0;
	elseif (t_1 <= 5e+295)
		tmp = Float64(1.0 * a_m);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(fma(99.0, k, -10.0) * a_m), k, a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 1e-315], t$95$0, If[LessEqual[t$95$1, 5e+295], N[(1.0 * a$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.999999985e-316 or 4.99999999999999991e295 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 97.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 36.1%

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

   if 9.999999985e-316 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.99999999999999991e295

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 69.0

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

   9. Applied rewrites 69.0%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 66.9%

       \[\leadsto 1 \cdot a \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 77.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{-315}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;1 \cdot a\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{+278}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m)))
   (*
    a_s
    (if (<= (/ t_0 (+ (* k k) (+ (* 10.0 k) 1.0))) 1e+278)
      (* (/ (pow k m) (fma (+ 10.0 k) k 1.0)) a_m)
      t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double tmp;
	if ((t_0 / ((k * k) + ((10.0 * k) + 1.0))) <= 1e+278) {
		tmp = (pow(k, m) / fma((10.0 + k), k, 1.0)) * a_m;
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(k * k) + Float64(Float64(10.0 * k) + 1.0))) <= 1e+278)
		tmp = Float64(Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)) * a_m);
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+278], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.99999999999999964e277

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.6

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 9.99999999999999964e277 < (/.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 51.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(10 \cdot k + 1\right)} \leq 10^{+278}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{a\_m}{\frac{\mathsf{fma}\left(10 + k, k, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 9.5e-6)
    (/ a_m (/ (fma (+ 10.0 k) k 1.0) (pow k m)))
    (* (pow k m) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 9.5e-6) {
		tmp = a_m / (fma((10.0 + k), k, 1.0) / pow(k, m));
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 9.5e-6)
		tmp = Float64(a_m / Float64(fma(Float64(10.0 + k), k, 1.0) / (k ^ m)));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 9.5e-6], N[(a$95$m / N[(N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 9.5000000000000005e-6

   1. Initial program 97.0%

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

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.0

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if 9.5000000000000005e-6 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(10 + k, k, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m)))
   (* a_s (if (<= m 9.5e-6) (/ t_0 (fma k 10.0 (fma k k 1.0))) t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double tmp;
	if (m <= 9.5e-6) {
		tmp = t_0 / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	tmp = 0.0
	if (m <= 9.5e-6)
		tmp = Float64(t_0 / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, 9.5e-6], N[(t$95$0 / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5000000000000005e-6

   1. Initial program 97.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if 9.5000000000000005e-6 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m}}\\ \mathbf{elif}\;m \leq 5.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -1.45e-11)
    (/ (pow k m) (/ 1.0 a_m))
    (if (<= m 5.1e-10) (/ a_m (fma (+ 10.0 k) k 1.0)) (* (pow k m) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -1.45e-11) {
		tmp = pow(k, m) / (1.0 / a_m);
	} else if (m <= 5.1e-10) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -1.45e-11)
		tmp = Float64((k ^ m) / Float64(1.0 / a_m));
	elseif (m <= 5.1e-10)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -1.45e-11], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.1e-10], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.45e-11

   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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-fma.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{1}}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if -1.45e-11 < m < 5.09999999999999997e-10

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 94.4%

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

   if 5.09999999999999997e-10 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 9: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m)))
   (*
    a_s
    (if (<= m -1.45e-11)
      t_0
      (if (<= m 5.1e-10) (/ a_m (fma (+ 10.0 k) k 1.0)) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double tmp;
	if (m <= -1.45e-11) {
		tmp = t_0;
	} else if (m <= 5.1e-10) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	tmp = 0.0
	if (m <= -1.45e-11)
		tmp = t_0;
	elseif (m <= 5.1e-10)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -1.45e-11], t$95$0, If[LessEqual[m, 5.1e-10], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.45e-11 or 5.09999999999999997e-10 < m

   1. Initial program 85.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.45e-11 < m < 5.09999999999999997e-10

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 94.4%

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

Alternative 10: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a\_m}{k}, \frac{99}{k} + -10, a\_m\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 128000000:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.21)
    (/ (fma (/ a_m k) (+ (/ 99.0 k) -10.0) a_m) (* k k))
    (if (<= m 128000000.0)
      (/ a_m (fma (+ 10.0 k) k 1.0))
      (fma (* (fma 99.0 k -10.0) a_m) k a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.21) {
		tmp = fma((a_m / k), ((99.0 / k) + -10.0), a_m) / (k * k);
	} else if (m <= 128000000.0) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((fma(99.0, k, -10.0) * a_m), k, a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.21)
		tmp = Float64(fma(Float64(a_m / k), Float64(Float64(99.0 / k) + -10.0), a_m) / Float64(k * k));
	elseif (m <= 128000000.0)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(fma(99.0, k, -10.0) * a_m), k, a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.21], N[(N[(N[(a$95$m / k), $MachinePrecision] * N[(N[(99.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] + a$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 128000000.0], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.209999999999999992

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 68.2%

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

   if -0.209999999999999992 < m < 1.28e8

   1. Initial program 95.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.5%

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

   if 1.28e8 < m

   1. Initial program 73.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 28.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{99}{k} + -10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 128000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 128000000:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.21)
    (/ a_m (* k k))
    (if (<= m 128000000.0)
      (/ a_m (fma (+ 10.0 k) k 1.0))
      (fma (* (fma 99.0 k -10.0) a_m) k a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.21) {
		tmp = a_m / (k * k);
	} else if (m <= 128000000.0) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((fma(99.0, k, -10.0) * a_m), k, a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.21)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 128000000.0)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(fma(99.0, k, -10.0) * a_m), k, a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.21], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 128000000.0], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.209999999999999992

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.2%

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

   if -0.209999999999999992 < m < 1.28e8

   1. Initial program 95.0%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.5%

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

   if 1.28e8 < m

   1. Initial program 73.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 28.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 128000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right) \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a\_m, a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k k))))
   (*
    a_s
    (if (<= k -1.35e-303)
      t_0
      (if (<= k 3.7e-9) (fma (* -10.0 k) a_m a_m) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double tmp;
	if (k <= -1.35e-303) {
		tmp = t_0;
	} else if (k <= 3.7e-9) {
		tmp = fma((-10.0 * k), a_m, a_m);
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * k))
	tmp = 0.0
	if (k <= -1.35e-303)
		tmp = t_0;
	elseif (k <= 3.7e-9)
		tmp = fma(Float64(-10.0 * k), a_m, a_m);
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -1.35e-303], t$95$0, If[LessEqual[k, 3.7e-9], N[(N[(-10.0 * k), $MachinePrecision] * a$95$m + a$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -1.34999999999999993e-303 or 3.7e-9 < k

   1. Initial program 82.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 45.8%

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

   if -1.34999999999999993e-303 < k < 3.7e-9

   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 51.7%

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

   10. Applied rewrites 51.7%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a, a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 25.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 92:\\ \;\;\;\;1 \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\_m\right) \cdot k\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 92.0) (* 1.0 a_m) (* (* -10.0 a_m) k))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 92.0) {
		tmp = 1.0 * a_m;
	} else {
		tmp = (-10.0 * a_m) * k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 92.0d0) then
        tmp = 1.0d0 * a_m
    else
        tmp = ((-10.0d0) * a_m) * k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 92.0) {
		tmp = 1.0 * a_m;
	} else {
		tmp = (-10.0 * a_m) * k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 92.0:
		tmp = 1.0 * a_m
	else:
		tmp = (-10.0 * a_m) * k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 92.0)
		tmp = Float64(1.0 * a_m);
	else
		tmp = Float64(Float64(-10.0 * a_m) * k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 92.0)
		tmp = 1.0 * a_m;
	else
		tmp = (-10.0 * a_m) * k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 92.0], N[(1.0 * a$95$m), $MachinePrecision], N[(N[(-10.0 * a$95$m), $MachinePrecision] * k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 92

   1. Initial program 97.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-fma.f64 N/A

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 73.2

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

   9. Applied rewrites 73.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 31.6%

       \[\leadsto 1 \cdot a \]

   if 92 < m

   1. Initial program 73.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 22.3%

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

   13. Applied rewrites 22.3%

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

Alternative 14: 20.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \left(1 \cdot a\_m\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (* 1.0 a_m)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * (1.0 * a_m);
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * (1.0d0 * a_m)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * (1.0 * a_m);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (1.0 * a_m)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(1.0 * a_m))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (1.0 * a_m);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(1.0 * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-*.f64 N/A

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

   14. +-commutative N/A

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

   15. +-commutative N/A

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

   16. associate-+r+ N/A

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

   17. lift-*.f64 N/A

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

   18. distribute-rgt-in N/A

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

   19. +-commutative N/A

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

   20. lift-+.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-fma.f64 N/A

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

6. Applied rewrites 87.2%

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

7. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 82.8

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

9. Applied rewrites 82.8%

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

10. Taylor expanded in m around 0

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

12. Applied rewrites 21.6%

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

Reproduce

herbie shell --seed 2024250 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))