Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.0%

Time: 10.8s

Alternatives: 17

Speedup: 1.1×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+217}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{{k}^{\left(-m\right)}}{a\_m} \cdot \left(k + 10\right), \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+217)
      (/ 1.0 (fma k (* (/ (pow k (- m)) a_m) (+ k 10.0)) (/ 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 = a_m * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+217) {
		tmp = 1.0 / fma(k, ((pow(k, -m) / a_m) * (k + 10.0)), (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(a_m * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+217)
		tmp = Float64(1.0 / fma(k, Float64(Float64((k ^ Float64(-m)) / a_m) * Float64(k + 10.0)), Float64(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[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+217], N[(1.0 / N[(k * N[(N[(N[Power[k, (-m)], $MachinePrecision] / a$95$m), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999996e216

   1. Initial program 93.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 93.7

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 93.7

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

   6. Applied rewrites 93.7%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 85.4%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 86.9%

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

   if 9.9999999999999996e216 < (/.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 50.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+217}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (pow k (- m))))
   (* a_s (/ 1.0 (fma k (/ (* k t_0) a_m) (/ t_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) {
	double t_0 = pow(k, -m);
	return a_s * (1.0 / fma(k, ((k * t_0) / a_m), (t_0 / a_m)));
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = k ^ Float64(-m)
	return Float64(a_s * Float64(1.0 / fma(k, Float64(Float64(k * t_0) / a_m), Float64(t_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_] := Block[{t$95$0 = N[Power[k, (-m)], $MachinePrecision]}, N[(a$95$s * N[(1.0 / N[(k * N[(N[(k * t$95$0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(t$95$0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 84.8

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-+.f64 84.8

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

4. Applied rewrites 84.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-neg.f64 84.9

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

6. Applied rewrites 84.9%

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

7. Taylor expanded in k around 0

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

   1. lower-fma.f64 N/A

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

9. Applied rewrites 88.4%

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

10. Taylor expanded in k around inf

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

12. Applied rewrites 99.0%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-19}:\\ \;\;\;\;{k}^{m} \cdot \frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{k \cdot \left(k \cdot {k}^{\left(-m\right)}\right)}\\ \end{array} \end{array} \]

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 -2.05e-19)
    (* (pow k m) (/ a_m (fma k (+ k 10.0) 1.0)))
    (if (<= m 2.1e-5)
      (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 1.0 a_m)))
      (/ a_m (* k (* k (pow k (- 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 <= -2.05e-19) {
		tmp = pow(k, m) * (a_m / fma(k, (k + 10.0), 1.0));
	} else if (m <= 2.1e-5) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (1.0 / a_m));
	} else {
		tmp = a_m / (k * (k * pow(k, -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 <= -2.05e-19)
		tmp = Float64((k ^ m) * Float64(a_m / fma(k, Float64(k + 10.0), 1.0)));
	elseif (m <= 2.1e-5)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(1.0 / a_m)));
	else
		tmp = Float64(a_m / Float64(k * Float64(k * (k ^ Float64(-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, -2.05e-19], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.1e-5], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(k * N[(k * N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.04999999999999993e-19

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -2.04999999999999993e-19 < m < 2.09999999999999988e-5

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 84.8

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 84.8

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.9%

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

   if 2.09999999999999988e-5 < m

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 74.0

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 74.0

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

   6. Applied rewrites 74.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 73.0%

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

   10. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. exp-to-pow N/A

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

       4. log-rec N/A

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

       5. distribute-lft-neg-in N/A

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

       6. *-commutative N/A

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

       7. mul-1-neg N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

       14. mul-1-neg N/A

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

       15. exp-to-pow N/A

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

       16. lower-pow.f64 N/A

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

       17. mul-1-neg N/A

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

       18. lower-neg.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-19}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k \cdot {k}^{\left(-m\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{a\_m \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{k \cdot \left(k \cdot {k}^{\left(-m\right)}\right)}\\ \end{array} \end{array} \]

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 -4e-7)
    (/ (* a_m (pow k m)) (fma 10.0 k 1.0))
    (if (<= m 2.1e-5)
      (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 1.0 a_m)))
      (/ a_m (* k (* k (pow k (- 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 <= -4e-7) {
		tmp = (a_m * pow(k, m)) / fma(10.0, k, 1.0);
	} else if (m <= 2.1e-5) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (1.0 / a_m));
	} else {
		tmp = a_m / (k * (k * pow(k, -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 <= -4e-7)
		tmp = Float64(Float64(a_m * (k ^ m)) / fma(10.0, k, 1.0));
	elseif (m <= 2.1e-5)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(1.0 / a_m)));
	else
		tmp = Float64(a_m / Float64(k * Float64(k * (k ^ Float64(-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, -4e-7], N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.1e-5], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(k * N[(k * N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -3.9999999999999998e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -3.9999999999999998e-7 < m < 2.09999999999999988e-5

   1. Initial program 85.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.3

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.3

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

   4. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.3

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

   6. Applied rewrites 85.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 98.0%

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

   if 2.09999999999999988e-5 < m

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 74.0

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 74.0

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

   6. Applied rewrites 74.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 73.0%

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

   10. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. exp-to-pow N/A

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

       4. log-rec N/A

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

       5. distribute-lft-neg-in N/A

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

       6. *-commutative N/A

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

       7. mul-1-neg N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. *-commutative N/A

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

       13. distribute-rgt-neg-in N/A

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

       14. mul-1-neg N/A

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

       15. exp-to-pow N/A

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

       16. lower-pow.f64 N/A

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

       17. mul-1-neg N/A

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

       18. lower-neg.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (pow k m))))
   (*
    a_s
    (if (<= m -4e-7)
      (/ t_0 (fma 10.0 k 1.0))
      (if (<= m 0.04) (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 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 = a_m * pow(k, m);
	double tmp;
	if (m <= -4e-7) {
		tmp = t_0 / fma(10.0, k, 1.0);
	} else if (m <= 0.04) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (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(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -4e-7)
		tmp = Float64(t_0 / fma(10.0, k, 1.0));
	elseif (m <= 0.04)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(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[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -4e-7], N[(t$95$0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.04], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -3.9999999999999998e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -3.9999999999999998e-7 < m < 0.0400000000000000008

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.1%

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

   if 0.0400000000000000008 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.000115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.04:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (pow k m))))
   (*
    a_s
    (if (<= m -0.000115)
      t_0
      (if (<= m 0.04) (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 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 = a_m * pow(k, m);
	double tmp;
	if (m <= -0.000115) {
		tmp = t_0;
	} else if (m <= 0.04) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (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(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -0.000115)
		tmp = t_0;
	elseif (m <= 0.04)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(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[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -0.000115], t$95$0, If[LessEqual[m, 0.04], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.15e-4 or 0.0400000000000000008 < m

   1. Initial program 84.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.15e-4 < m < 0.0400000000000000008

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.1%

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

Alternative 7: 70.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+174}:\\ \;\;\;\;\frac{a\_m + \frac{\frac{\mathsf{fma}\left(a\_m, \frac{10}{k}, a\_m \cdot 99\right)}{k} + a\_m \cdot -10}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq -0.31:\\ \;\;\;\;\frac{\frac{a\_m \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -1.1e+174)
    (/
     (+ a_m (/ (+ (/ (fma a_m (/ 10.0 k) (* a_m 99.0)) k) (* a_m -10.0)) k))
     (* k k))
    (if (<= m -0.31)
      (/ (/ (* a_m -980.0) (* k (* k k))) (* k k))
      (if (<= m 0.9)
        (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 1.0 a_m)))
        (* k (* k (* a_m 99.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -1.1e+174) {
		tmp = (a_m + (((fma(a_m, (10.0 / k), (a_m * 99.0)) / k) + (a_m * -10.0)) / k)) / (k * k);
	} else if (m <= -0.31) {
		tmp = ((a_m * -980.0) / (k * (k * k))) / (k * k);
	} else if (m <= 0.9) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (1.0 / a_m));
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -1.1e+174)
		tmp = Float64(Float64(a_m + Float64(Float64(Float64(fma(a_m, Float64(10.0 / k), Float64(a_m * 99.0)) / k) + Float64(a_m * -10.0)) / k)) / Float64(k * k));
	elseif (m <= -0.31)
		tmp = Float64(Float64(Float64(a_m * -980.0) / Float64(k * Float64(k * k))) / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(1.0 / a_m)));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -1.1e+174], N[(N[(a$95$m + N[(N[(N[(N[(a$95$m * N[(10.0 / k), $MachinePrecision] + N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] + N[(a$95$m * -10.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -0.31], N[(N[(N[(a$95$m * -980.0), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -1.1000000000000001e174

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 41.2%

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, \mathsf{fma}\left(-990, \frac{a}{k}, a \cdot 99\right)\right)}{k} + a \cdot -10}{k}}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, 99 \cdot a\right)}{k} + a \cdot -10}{k}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.8%

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

   if -1.1000000000000001e174 < m < -0.309999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 55.2%

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, \mathsf{fma}\left(-990, \frac{a}{k}, a \cdot 99\right)\right)}{k} + a \cdot -10}{k}}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{-990 \cdot a + 10 \cdot a}{{k}^{3}}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{\frac{a \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k} \]

   if -0.309999999999999998 < m < 0.900000000000000022

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.1%

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

   if 0.900000000000000022 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.7%

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

Alternative 8: 70.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{a\_m + \frac{\mathsf{fma}\left(\frac{a\_m}{k}, 99, a\_m \cdot -10\right)}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq -0.31:\\ \;\;\;\;\frac{\frac{a\_m \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -3.9e+173)
    (/ (+ a_m (/ (fma (/ a_m k) 99.0 (* a_m -10.0)) k)) (* k k))
    (if (<= m -0.31)
      (/ (/ (* a_m -980.0) (* k (* k k))) (* k k))
      (if (<= m 0.9)
        (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 1.0 a_m)))
        (* k (* k (* a_m 99.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -3.9e+173) {
		tmp = (a_m + (fma((a_m / k), 99.0, (a_m * -10.0)) / k)) / (k * k);
	} else if (m <= -0.31) {
		tmp = ((a_m * -980.0) / (k * (k * k))) / (k * k);
	} else if (m <= 0.9) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (1.0 / a_m));
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -3.9e+173)
		tmp = Float64(Float64(a_m + Float64(fma(Float64(a_m / k), 99.0, Float64(a_m * -10.0)) / k)) / Float64(k * k));
	elseif (m <= -0.31)
		tmp = Float64(Float64(Float64(a_m * -980.0) / Float64(k * Float64(k * k))) / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(1.0 / a_m)));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -3.9e+173], N[(N[(a$95$m + N[(N[(N[(a$95$m / k), $MachinePrecision] * 99.0 + N[(a$95$m * -10.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -0.31], N[(N[(N[(a$95$m * -980.0), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -3.8999999999999998e173

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.4%

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

   if -3.8999999999999998e173 < m < -0.309999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 55.2%

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, \mathsf{fma}\left(-990, \frac{a}{k}, a \cdot 99\right)\right)}{k} + a \cdot -10}{k}}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{-990 \cdot a + 10 \cdot a}{{k}^{3}}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{\frac{a \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k} \]

   if -0.309999999999999998 < m < 0.900000000000000022

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.1%

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

   if 0.900000000000000022 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.7%

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

Alternative 9: 70.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq -0.31:\\ \;\;\;\;\frac{\frac{a\_m \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k + 10}{a\_m}, \frac{1}{a\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -3.9e+173)
    (/ a_m (* k k))
    (if (<= m -0.31)
      (/ (/ (* a_m -980.0) (* k (* k k))) (* k k))
      (if (<= m 0.9)
        (/ 1.0 (fma k (/ (+ k 10.0) a_m) (/ 1.0 a_m)))
        (* k (* k (* a_m 99.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -3.9e+173) {
		tmp = a_m / (k * k);
	} else if (m <= -0.31) {
		tmp = ((a_m * -980.0) / (k * (k * k))) / (k * k);
	} else if (m <= 0.9) {
		tmp = 1.0 / fma(k, ((k + 10.0) / a_m), (1.0 / a_m));
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -3.9e+173)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= -0.31)
		tmp = Float64(Float64(Float64(a_m * -980.0) / Float64(k * Float64(k * k))) / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k + 10.0) / a_m), Float64(1.0 / a_m)));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -3.9e+173], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -0.31], N[(N[(N[(a$95$m * -980.0), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(1.0 / N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision] + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -3.8999999999999998e173

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.0%

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

   if -3.8999999999999998e173 < m < -0.309999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 55.2%

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, \mathsf{fma}\left(-990, \frac{a}{k}, a \cdot 99\right)\right)}{k} + a \cdot -10}{k}}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{-990 \cdot a + 10 \cdot a}{{k}^{3}}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{\frac{a \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k} \]

   if -0.309999999999999998 < m < 0.900000000000000022

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-neg.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 97.1%

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

   if 0.900000000000000022 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.7%

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

Alternative 10: 68.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq -4.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{a\_m \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -3.9e+173)
    (/ a_m (* k k))
    (if (<= m -4.1e+21)
      (/ (/ (* a_m -980.0) (* k (* k k))) (* k k))
      (if (<= m 0.9)
        (/ a_m (fma k (+ k 10.0) 1.0))
        (* k (* k (* a_m 99.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -3.9e+173) {
		tmp = a_m / (k * k);
	} else if (m <= -4.1e+21) {
		tmp = ((a_m * -980.0) / (k * (k * k))) / (k * k);
	} else if (m <= 0.9) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -3.9e+173)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= -4.1e+21)
		tmp = Float64(Float64(Float64(a_m * -980.0) / Float64(k * Float64(k * k))) / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -3.9e+173], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -4.1e+21], N[(N[(N[(a$95$m * -980.0), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -3.8999999999999998e173

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.0%

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

   if -3.8999999999999998e173 < m < -4.1e21

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 52.7%

      \[\leadsto \frac{a + \frac{\frac{\mathsf{fma}\left(a, \frac{10}{k}, \mathsf{fma}\left(-990, \frac{a}{k}, a \cdot 99\right)\right)}{k} + a \cdot -10}{k}}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{-990 \cdot a + 10 \cdot a}{{k}^{3}}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.2%

       \[\leadsto \frac{\frac{a \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k} \]

   if -4.1e21 < m < 0.900000000000000022

   1. Initial program 86.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if 0.900000000000000022 < m

   1. Initial program 73.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq -4.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{a \cdot -980}{k \cdot \left(k \cdot k\right)}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -4.2e+38)
    (/ a_m (* k k))
    (if (<= m 0.9) (/ a_m (fma k (+ k 10.0) 1.0)) (* k (* k (* a_m 99.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.2e+38) {
		tmp = a_m / (k * k);
	} else if (m <= 0.9) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -4.2e+38)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -4.2e+38], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.2e38

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 43.7

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.4%

      \[\leadsto \frac{a}{k \cdot \color{blue}{k}} \]

   if -4.2e38 < m < 0.900000000000000022

   1. Initial program 86.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 83.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   if 0.900000000000000022 < m

   1. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -7.2e-188)
    (/ a_m (* k k))
    (if (<= m 0.9) (/ a_m (fma k 10.0 1.0)) (* k (* k (* a_m 99.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -7.2e-188) {
		tmp = a_m / (k * k);
	} else if (m <= 0.9) {
		tmp = a_m / fma(k, 10.0, 1.0);
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -7.2e-188)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.9)
		tmp = Float64(a_m / fma(k, 10.0, 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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_] := N[(a$95$s * If[LessEqual[m, -7.2e-188], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(a$95$m / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -7.1999999999999994e-188

   1. Initial program 96.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 57.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.4%

      \[\leadsto \frac{a}{k \cdot \color{blue}{k}} \]

   if -7.1999999999999994e-188 < m < 0.900000000000000022

   1. Initial program 85.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 82.8

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.3%

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \]

   if 0.900000000000000022 < m

   1. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.6:\\ \;\;\;\;a\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -2.1e-191)
    (/ a_m (* k k))
    (if (<= m 0.6) (* a_m 1.0) (* k (* k (* a_m 99.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -2.1e-191) {
		tmp = a_m / (k * k);
	} else if (m <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= (-2.1d-191)) then
        tmp = a_m / (k * k)
    else if (m <= 0.6d0) then
        tmp = a_m * 1.0d0
    else
        tmp = k * (k * (a_m * 99.0d0))
    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 <= -2.1e-191) {
		tmp = a_m / (k * k);
	} else if (m <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -2.1e-191:
		tmp = a_m / (k * k)
	elif m <= 0.6:
		tmp = a_m * 1.0
	else:
		tmp = k * (k * (a_m * 99.0))
	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 <= -2.1e-191)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.6)
		tmp = Float64(a_m * 1.0);
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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 <= -2.1e-191)
		tmp = a_m / (k * k);
	elseif (m <= 0.6)
		tmp = a_m * 1.0;
	else
		tmp = k * (k * (a_m * 99.0));
	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, -2.1e-191], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.6], N[(a$95$m * 1.0), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.09999999999999985e-191

   1. Initial program 96.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 57.2

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.4%

      \[\leadsto \frac{a}{k \cdot \color{blue}{k}} \]

   if -2.09999999999999985e-191 < m < 0.599999999999999978

   1. Initial program 85.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right) \cdot \frac{1}{a \cdot {k}^{m}}}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}{\frac{1}{a \cdot {k}^{m}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\frac{1}{a \cdot {k}^{m}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\left(1 + 10 \cdot k\right) - k \cdot k}}}}{\frac{1}{a \cdot {k}^{m}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}{\frac{1}{a \cdot {k}^{m}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 52.9

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   7. Applied rewrites 52.9%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   8. Taylor expanded in m around 0

      \[\leadsto a \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 52.0%

       \[\leadsto a \cdot 1 \]

   if 0.599999999999999978 < m

   1. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 41.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{a\_m}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.6:\\ \;\;\;\;a\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 -1e-16)
    (/ a_m (* k 10.0))
    (if (<= m 0.6) (* a_m 1.0) (* k (* k (* a_m 99.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -1e-16) {
		tmp = a_m / (k * 10.0);
	} else if (m <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= (-1d-16)) then
        tmp = a_m / (k * 10.0d0)
    else if (m <= 0.6d0) then
        tmp = a_m * 1.0d0
    else
        tmp = k * (k * (a_m * 99.0d0))
    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 <= -1e-16) {
		tmp = a_m / (k * 10.0);
	} else if (m <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= -1e-16:
		tmp = a_m / (k * 10.0)
	elif m <= 0.6:
		tmp = a_m * 1.0
	else:
		tmp = k * (k * (a_m * 99.0))
	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 <= -1e-16)
		tmp = Float64(a_m / Float64(k * 10.0));
	elseif (m <= 0.6)
		tmp = Float64(a_m * 1.0);
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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 <= -1e-16)
		tmp = a_m / (k * 10.0);
	elseif (m <= 0.6)
		tmp = a_m * 1.0;
	else
		tmp = k * (k * (a_m * 99.0));
	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, -1e-16], N[(a$95$m / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.6], N[(a$95$m * 1.0), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -9.9999999999999998e-17

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 47.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{{k}^{2} \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.5%

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k + 10\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{a}{10 \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 22.7%

       \[\leadsto \frac{a}{k \cdot 10} \]

   if -9.9999999999999998e-17 < m < 0.599999999999999978

   1. Initial program 85.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right) \cdot \frac{1}{a \cdot {k}^{m}}}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}{\frac{1}{a \cdot {k}^{m}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\frac{1}{a \cdot {k}^{m}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\left(1 + 10 \cdot k\right) - k \cdot k}}}}{\frac{1}{a \cdot {k}^{m}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}{\frac{1}{a \cdot {k}^{m}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 50.4

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   7. Applied rewrites 50.4%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   8. Taylor expanded in m around 0

      \[\leadsto a \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 49.7%

       \[\leadsto a \cdot 1 \]

   if 0.599999999999999978 < m

   1. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 35.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 0.6:\\ \;\;\;\;a\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a\_m \cdot 99\right)\right)\\ \end{array} \end{array} \]

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 0.6) (* a_m 1.0) (* k (* k (* a_m 99.0))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= 0.6d0) then
        tmp = a_m * 1.0d0
    else
        tmp = k * (k * (a_m * 99.0d0))
    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 <= 0.6) {
		tmp = a_m * 1.0;
	} else {
		tmp = k * (k * (a_m * 99.0));
	}
	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 <= 0.6:
		tmp = a_m * 1.0
	else:
		tmp = k * (k * (a_m * 99.0))
	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 <= 0.6)
		tmp = Float64(a_m * 1.0);
	else
		tmp = Float64(k * Float64(k * Float64(a_m * 99.0)));
	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 <= 0.6)
		tmp = a_m * 1.0;
	else
		tmp = k * (k * (a_m * 99.0));
	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, 0.6], N[(a$95$m * 1.0), $MachinePrecision], N[(k * N[(k * N[(a$95$m * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 0.599999999999999978

   1. Initial program 91.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right) \cdot \frac{1}{a \cdot {k}^{m}}}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}{\frac{1}{a \cdot {k}^{m}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\frac{1}{a \cdot {k}^{m}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\left(1 + 10 \cdot k\right) - k \cdot k}}}}{\frac{1}{a \cdot {k}^{m}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}{\frac{1}{a \cdot {k}^{m}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 70.9

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   7. Applied rewrites 70.9%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   8. Taylor expanded in m around 0

      \[\leadsto a \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 29.1%

       \[\leadsto a \cdot 1 \]

   if 0.599999999999999978 < m

   1. Initial program 73.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, a \cdot 99, a \cdot -10\right)}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.7%

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 24.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 6.4 \cdot 10^{+22}:\\ \;\;\;\;a\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 6.4e+22) (* a_m 1.0) (* a_m (* k -10.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 6.4e+22) {
		tmp = a_m * 1.0;
	} else {
		tmp = a_m * (k * -10.0);
	}
	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 <= 6.4d+22) then
        tmp = a_m * 1.0d0
    else
        tmp = a_m * (k * (-10.0d0))
    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 <= 6.4e+22) {
		tmp = a_m * 1.0;
	} else {
		tmp = a_m * (k * -10.0);
	}
	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 <= 6.4e+22:
		tmp = a_m * 1.0
	else:
		tmp = a_m * (k * -10.0)
	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 <= 6.4e+22)
		tmp = Float64(a_m * 1.0);
	else
		tmp = Float64(a_m * Float64(k * -10.0));
	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 <= 6.4e+22)
		tmp = a_m * 1.0;
	else
		tmp = a_m * (k * -10.0);
	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, 6.4e+22], N[(a$95$m * 1.0), $MachinePrecision], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 6.4e22

   1. Initial program 91.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right) \cdot \frac{1}{a \cdot {k}^{m}}}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}{\frac{1}{a \cdot {k}^{m}}}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\frac{1}{a \cdot {k}^{m}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\left(1 + 10 \cdot k\right) - k \cdot k}}}}{\frac{1}{a \cdot {k}^{m}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}{\frac{1}{a \cdot {k}^{m}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 72.1

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   7. Applied rewrites 72.1%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   8. Taylor expanded in m around 0

      \[\leadsto a \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 28.0%

       \[\leadsto a \cdot 1 \]

   if 6.4e22 < 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. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 2.9

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 12.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto -10 \cdot \left(a \cdot \color{blue}{k}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 26.2%

       \[\leadsto a \cdot \left(k \cdot \color{blue}{-10}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 19.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \left(a\_m \cdot 1\right) \end{array} \]

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 (* a_m 1.0)))

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 * (a_m * 1.0);
}

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 * (a_m * 1.0d0)
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 * (a_m * 1.0);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (a_m * 1.0)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m * 1.0))
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 * (a_m * 1.0);
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[(a$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{a \cdot {k}^{m}}}} \]

   3. div-inv N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + 10 \cdot k\right) + k \cdot k\right) \cdot \frac{1}{a \cdot {k}^{m}}}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}{\frac{1}{a \cdot {k}^{m}}}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}}}{\frac{1}{a \cdot {k}^{m}}} \]

   6. flip-+ N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\left(1 + 10 \cdot k\right) - k \cdot k}}}}{\frac{1}{a \cdot {k}^{m}}} \]

   7. clear-num N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}{\frac{1}{a \cdot {k}^{m}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(1 + 10 \cdot k\right) - k \cdot k}{\left(1 + 10 \cdot k\right) \cdot \left(1 + 10 \cdot k\right) - \left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

4. Applied rewrites 84.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}}{\frac{1}{a \cdot {k}^{m}}}} \]

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   2. lower-pow.f64 82.0

      \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

7. Applied rewrites 82.0%

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

8. Taylor expanded in m around 0

   \[\leadsto a \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 19.4%

    \[\leadsto a \cdot 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))