Falkner and Boettcher, Appendix A

Percentage Accurate: 89.6% → 97.3%

Time: 12.8s

Alternatives: 16

Speedup: 1.1×

• 21.7% 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: 89.6% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+181}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+181)
      (/ t_0 (fma (+ k 10.0) k 1.0))
      (/ a_m (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 t_0 = a_m * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+181) {
		tmp = t_0 / fma((k + 10.0), k, 1.0);
	} else {
		tmp = a_m / 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)
	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+181)
		tmp = Float64(t_0 / fma(Float64(k + 10.0), k, 1.0));
	else
		tmp = Float64(a_m / (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_] := 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+181], N[(t$95$0 / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999992e180

   1. Initial program 97.6%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   if 9.9999999999999992e180 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 70.2%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. neg-lowering-neg.f64 75.4

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 100.0

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{a}{{k}^{\left(-m\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+181}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+181}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+181)
    (* a_m (/ (pow k m) (fma k (+ k 10.0) 1.0)))
    (/ a_m (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 (((a_m * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+181) {
		tmp = a_m * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a_m / 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 (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+181)
		tmp = Float64(a_m * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a_m / (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[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+181], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999992e180

   1. Initial program 97.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-lowering-+.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   if 9.9999999999999992e180 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 70.2%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. neg-lowering-neg.f64 75.4

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 100.0

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{a}{{k}^{\left(-m\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+181}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 30.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, k \cdot -10, a\_m\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 5e-310)
    (- (* a_m (* k k)))
    (fma a_m (* k -10.0) 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 tmp;
	if (((a_m * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 5e-310) {
		tmp = -(a_m * (k * k));
	} else {
		tmp = fma(a_m, (k * -10.0), a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 5e-310)
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	else
		tmp = fma(a_m, Float64(k * -10.0), a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-310], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), N[(a$95$m * N[(k * -10.0), $MachinePrecision] + a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.999999999999985e-310

   1. Initial program 97.3%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 55.0

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

   5. Simplified 55.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 33.5

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

   10. Simplified 33.5%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 17.7

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

   13. Simplified 17.7%

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

   if 4.999999999999985e-310 < (/.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 79.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 34.8

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

   5. Simplified 34.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 33.5

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

   8. Simplified 33.5%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)} \cdot \mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -3.6e-13)
    (/ a_m (* (pow k (- m)) (fma k 10.0 1.0)))
    (if (<= m 1.9e-8) (/ a_m (fma k (+ k 10.0) 1.0)) (* a_m (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 <= -3.6e-13) {
		tmp = a_m / (pow(k, -m) * fma(k, 10.0, 1.0));
	} else if (m <= 1.9e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a_m * 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 <= -3.6e-13)
		tmp = Float64(a_m / Float64((k ^ Float64(-m)) * fma(k, 10.0, 1.0)));
	elseif (m <= 1.9e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a_m * (k ^ 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, -3.6e-13], N[(a$95$m / N[(N[Power[k, (-m)], $MachinePrecision] * N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -3.5999999999999998e-13

   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. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. neg-lowering-neg.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   if -3.5999999999999998e-13 < m < 1.90000000000000014e-8

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 94.1

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

   5. Simplified 94.1%

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

   if 1.90000000000000014e-8 < m

   1. Initial program 79.5%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)} \cdot \mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= m -6.8e-10)
      (/ t_0 (fma 10.0 k 1.0))
      (if (<= m 1.05e-8) (/ a_m (fma k (+ k 10.0) 1.0)) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= -6.8e-10) {
		tmp = t_0 / fma(10.0, k, 1.0);
	} else if (m <= 1.05e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -6.8e-10)
		tmp = Float64(t_0 / fma(10.0, k, 1.0));
	elseif (m <= 1.05e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -6.8e-10], N[(t$95$0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -6.8000000000000003e-10

   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. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

   if -6.8000000000000003e-10 < m < 1.04999999999999997e-8

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 94.1

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

   5. Simplified 94.1%

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

   if 1.04999999999999997e-8 < m

   1. Initial program 79.5%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -1.6e-10)
    (/ a_m (pow k (- m)))
    (if (<= m 2.9e-8) (/ a_m (fma k (+ k 10.0) 1.0)) (* a_m (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 <= -1.6e-10) {
		tmp = a_m / pow(k, -m);
	} else if (m <= 2.9e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a_m * 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 <= -1.6e-10)
		tmp = Float64(a_m / (k ^ Float64(-m)));
	elseif (m <= 2.9e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a_m * (k ^ m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -1.6e-10], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.9e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.5999999999999999e-10

   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. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. neg-lowering-neg.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 98.9

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

   10. Simplified 98.9%

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

   if -1.5999999999999999e-10 < m < 2.9000000000000002e-8

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 94.1

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

   5. Simplified 94.1%

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

   if 2.9000000000000002e-8 < m

   1. Initial program 79.5%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= m -1.7e-9)
      t_0
      (if (<= m 2e-8) (/ a_m (fma k (+ k 10.0) 1.0)) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= -1.7e-9) {
		tmp = t_0;
	} else if (m <= 2e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -1.7e-9)
		tmp = t_0;
	elseif (m <= 2e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -1.7e-9], t$95$0, If[LessEqual[m, 2e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.6999999999999999e-9 or 2e-8 < m

   1. Initial program 89.9%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 99.4

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

   5. Simplified 99.4%

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

   if -1.6999999999999999e-9 < m < 2e-8

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 94.1

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

   5. Simplified 94.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a\_m \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 0.0155:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a\_m \cdot \mathsf{fma}\left(-20, k, -100\right), -a\_m\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -355.0)
    (/ (* a_m 99.0) (* (* k k) (* k k)))
    (if (<= m 0.0155)
      (/ a_m (fma k (+ k 10.0) 1.0))
      (* (* k k) (fma (* k k) (* a_m (fma -20.0 k -100.0)) (- 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 tmp;
	if (m <= -355.0) {
		tmp = (a_m * 99.0) / ((k * k) * (k * k));
	} else if (m <= 0.0155) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (k * k) * fma((k * k), (a_m * fma(-20.0, k, -100.0)), -a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -355.0)
		tmp = Float64(Float64(a_m * 99.0) / Float64(Float64(k * k) * Float64(k * k)));
	elseif (m <= 0.0155)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(k * k) * fma(Float64(k * k), Float64(a_m * fma(-20.0, k, -100.0)), Float64(-a_m)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -355.0], N[(N[(a$95$m * 99.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0155], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(a$95$m * N[(-20.0 * k + -100.0), $MachinePrecision]), $MachinePrecision] + (-a$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -355

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 45.7

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

   7. Applied egg-rr 45.7%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 69.6%

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

   11. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{a \cdot 99}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

       6. pow-sqr N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       8. unpow2 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       10. unpow2 N/A

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

       11. *-lowering-*.f64 79.0

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

   13. Simplified 79.0%

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

   if -355 < m < 0.0155

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 91.5

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

   5. Simplified 91.5%

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

   if 0.0155 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.4

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

   5. Simplified 3.4%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 3.0%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 34.7

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

   10. Simplified 34.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + {k}^{2} \cdot \left(-20 \cdot \left(a \cdot k\right) - 100 \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(-1 \cdot a + {k}^{2} \cdot \left(-20 \cdot \left(a \cdot k\right) - 100 \cdot a\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + {k}^{2} \cdot \left(-20 \cdot \left(a \cdot k\right) - 100 \cdot a\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(-1 \cdot a + {k}^{2} \cdot \left(-20 \cdot \left(a \cdot k\right) - 100 \cdot a\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(-20 \cdot \left(a \cdot k\right) - 100 \cdot a\right) + -1 \cdot a\right)} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, -20 \cdot \left(a \cdot k\right) - 100 \cdot a, -1 \cdot a\right)} \]

       6. unpow2 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, -20 \cdot \left(a \cdot k\right) - 100 \cdot a, -1 \cdot a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, -20 \cdot \left(a \cdot k\right) - 100 \cdot a, -1 \cdot a\right) \]

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

          \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{-20 \cdot \left(a \cdot k\right) + \left(\mathsf{neg}\left(100\right)\right) \cdot a}, -1 \cdot a\right) \]

       9. *-commutative N/A

          \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, -20 \cdot \color{blue}{\left(k \cdot a\right)} + \left(\mathsf{neg}\left(100\right)\right) \cdot a, -1 \cdot a\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\left(-20 \cdot k\right) \cdot a} + \left(\mathsf{neg}\left(100\right)\right) \cdot a, -1 \cdot a\right) \]

       11. metadata-eval N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \left(-20 \cdot k\right) \cdot a + \color{blue}{-100} \cdot a, -1 \cdot a\right) \]

       12. distribute-rgt-out N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot \left(-20 \cdot k + -100\right)}, -1 \cdot a\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot \left(-20 \cdot k + -100\right)}, -1 \cdot a\right) \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a \cdot \color{blue}{\mathsf{fma}\left(-20, k, -100\right)}, -1 \cdot a\right) \]

       15. mul-1-neg N/A

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a \cdot \mathsf{fma}\left(-20, k, -100\right), \color{blue}{\mathsf{neg}\left(a\right)}\right) \]

       16. neg-lowering-neg.f64 52.5

           \[\leadsto \left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a \cdot \mathsf{fma}\left(-20, k, -100\right), \color{blue}{-a}\right) \]

   13. Simplified 52.5%

       \[\leadsto \color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a \cdot \mathsf{fma}\left(-20, k, -100\right), -a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 0.0155:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, a \cdot \mathsf{fma}\left(-20, k, -100\right), -a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a\_m \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{+61}:\\ \;\;\;\;a\_m \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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 -355.0)
    (/ (* a_m 99.0) (* (* k k) (* k k)))
    (if (<= m 4.2e-8)
      (/ a_m (fma k (+ k 10.0) 1.0))
      (if (<= m 2.3e+61)
        (* a_m (fma k (fma k 99.0 -10.0) 1.0))
        (- (* a_m (* k k))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -355.0) {
		tmp = (a_m * 99.0) / ((k * k) * (k * k));
	} else if (m <= 4.2e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else if (m <= 2.3e+61) {
		tmp = a_m * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -355.0)
		tmp = Float64(Float64(a_m * 99.0) / Float64(Float64(k * k) * Float64(k * k)));
	elseif (m <= 4.2e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	elseif (m <= 2.3e+61)
		tmp = Float64(a_m * fma(k, fma(k, 99.0, -10.0), 1.0));
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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, -355.0], N[(N[(a$95$m * 99.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3e+61], N[(a$95$m * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -355

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 45.7

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

   7. Applied egg-rr 45.7%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 69.6%

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

   11. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{a \cdot 99}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

       6. pow-sqr N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       8. unpow2 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       10. unpow2 N/A

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

       11. *-lowering-*.f64 79.0

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

   13. Simplified 79.0%

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

   if -355 < m < 4.19999999999999989e-8

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 92.1

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

   5. Simplified 92.1%

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

   if 4.19999999999999989e-8 < m < 2.3e61

   1. Initial program 62.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 6.1

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

   5. Simplified 6.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 6.1

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

   7. Applied egg-rr 6.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. accelerator-lowering-fma.f64 55.4

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]

   if 2.3e61 < m

   1. Initial program 83.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.8

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

   10. Simplified 38.8%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 38.3

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

   13. Simplified 38.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{+61}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+61}:\\ \;\;\;\;a\_m \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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 -355.0)
    (/ a_m (* k k))
    (if (<= m 4.2e-8)
      (/ a_m (fma k (+ k 10.0) 1.0))
      (if (<= m 4.6e+61)
        (* a_m (fma k (fma k 99.0 -10.0) 1.0))
        (- (* a_m (* k k))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -355.0) {
		tmp = a_m / (k * k);
	} else if (m <= 4.2e-8) {
		tmp = a_m / fma(k, (k + 10.0), 1.0);
	} else if (m <= 4.6e+61) {
		tmp = a_m * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -355.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 4.2e-8)
		tmp = Float64(a_m / fma(k, Float64(k + 10.0), 1.0));
	elseif (m <= 4.6e+61)
		tmp = Float64(a_m * fma(k, fma(k, 99.0, -10.0), 1.0));
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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, -355.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-8], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.6e+61], N[(a$95$m * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -355

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

   6. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

   if -355 < m < 4.19999999999999989e-8

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 92.1

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

   5. Simplified 92.1%

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

   if 4.19999999999999989e-8 < m < 4.5999999999999999e61

   1. Initial program 62.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 6.1

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

   5. Simplified 6.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 6.1

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

   7. Applied egg-rr 6.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. accelerator-lowering-fma.f64 55.4

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]

   if 4.5999999999999999e61 < m

   1. Initial program 83.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.8

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

   10. Simplified 38.8%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 38.3

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

   13. Simplified 38.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+61}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{+61}:\\ \;\;\;\;a\_m \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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 -355.0)
    (/ a_m (* k k))
    (if (<= m 4.2e-8)
      (/ a_m (fma k k 1.0))
      (if (<= m 1.18e+61)
        (* a_m (fma k (fma k 99.0 -10.0) 1.0))
        (- (* a_m (* k k))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -355.0) {
		tmp = a_m / (k * k);
	} else if (m <= 4.2e-8) {
		tmp = a_m / fma(k, k, 1.0);
	} else if (m <= 1.18e+61) {
		tmp = a_m * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -355.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 4.2e-8)
		tmp = Float64(a_m / fma(k, k, 1.0));
	elseif (m <= 1.18e+61)
		tmp = Float64(a_m * fma(k, fma(k, 99.0, -10.0), 1.0));
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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, -355.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-8], N[(a$95$m / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.18e+61], N[(a$95$m * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -355

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

   6. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 63.7

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

   8. Simplified 63.7%

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

   if -355 < m < 4.19999999999999989e-8

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in k around inf

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

   8. Simplified 88.4%

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

   if 4.19999999999999989e-8 < m < 1.18000000000000004e61

   1. Initial program 62.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 6.1

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

   5. Simplified 6.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 6.1

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

   7. Applied egg-rr 6.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. accelerator-lowering-fma.f64 55.4

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]

   if 1.18000000000000004e61 < m

   1. Initial program 83.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.8

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

   10. Simplified 38.8%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 38.3

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

   13. Simplified 38.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -355:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{+61}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{+61}:\\ \;\;\;\;a\_m \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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-50)
    (/ a_m (* k k))
    (if (<= m 4.2e-8)
      (/ a_m (fma k 10.0 1.0))
      (if (<= m 1.02e+61)
        (* a_m (fma k (fma k 99.0 -10.0) 1.0))
        (- (* a_m (* k k))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.2e-50) {
		tmp = a_m / (k * k);
	} else if (m <= 4.2e-8) {
		tmp = a_m / fma(k, 10.0, 1.0);
	} else if (m <= 1.02e+61) {
		tmp = a_m * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -4.2e-50)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 4.2e-8)
		tmp = Float64(a_m / fma(k, 10.0, 1.0));
	elseif (m <= 1.02e+61)
		tmp = Float64(a_m * fma(k, fma(k, 99.0, -10.0), 1.0));
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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-50], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-8], N[(a$95$m / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.02e+61], N[(a$95$m * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -4.2000000000000002e-50

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 50.5

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

   5. Simplified 50.5%

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

   6. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if -4.2000000000000002e-50 < m < 4.19999999999999989e-8

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 68.6

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

   8. Simplified 68.6%

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

   if 4.19999999999999989e-8 < m < 1.01999999999999999e61

   1. Initial program 62.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 6.1

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

   5. Simplified 6.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 6.1

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

   7. Applied egg-rr 6.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. accelerator-lowering-fma.f64 55.4

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]

   if 1.01999999999999999e61 < m

   1. Initial program 83.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.8

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

   10. Simplified 38.8%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 38.3

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

   13. Simplified 38.3%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 1.02 \cdot 10^{+61}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -2.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;a\_m \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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 -2.2e-57)
    (/ a_m (* k k))
    (if (<= m 3.5e+61)
      (* a_m (fma k (fma k 99.0 -10.0) 1.0))
      (- (* a_m (* k k)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -2.2e-57) {
		tmp = a_m / (k * k);
	} else if (m <= 3.5e+61) {
		tmp = a_m * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -2.2e-57)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 3.5e+61)
		tmp = Float64(a_m * fma(k, fma(k, 99.0, -10.0), 1.0));
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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, -2.2e-57], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.5e+61], N[(a$95$m * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.19999999999999999e-57

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 51.5

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

   5. Simplified 51.5%

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

   6. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 62.2

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

   8. Simplified 62.2%

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

   if -2.19999999999999999e-57 < m < 3.50000000000000018e61

   1. Initial program 88.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 78.3

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

   5. Simplified 78.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 78.3

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

   7. Applied egg-rr 78.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. accelerator-lowering-fma.f64 57.4

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Simplified 57.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]

   if 3.50000000000000018e61 < m

   1. Initial program 83.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.3

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

   5. Simplified 3.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.8

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

   10. Simplified 38.8%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 38.3

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

   13. Simplified 38.3%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -6.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, k \cdot -10, a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-a\_m \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

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 -6.9e-62)
    (/ a_m (* k k))
    (if (<= m 2e+22) (fma a_m (* k -10.0) a_m) (- (* a_m (* k k)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -6.9e-62) {
		tmp = a_m / (k * k);
	} else if (m <= 2e+22) {
		tmp = fma(a_m, (k * -10.0), a_m);
	} else {
		tmp = -(a_m * (k * k));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -6.9e-62)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 2e+22)
		tmp = fma(a_m, Float64(k * -10.0), a_m);
	else
		tmp = Float64(-Float64(a_m * Float64(k * k)));
	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, -6.9e-62], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2e+22], N[(a$95$m * N[(k * -10.0), $MachinePrecision] + a$95$m), $MachinePrecision], (-N[(a$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -6.89999999999999958e-62

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 51.5

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

   5. Simplified 51.5%

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

   6. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 62.2

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

   8. Simplified 62.2%

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

   if -6.89999999999999958e-62 < m < 2e22

   1. Initial program 92.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 86.4

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

   5. Simplified 86.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 54.3

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

   8. Simplified 54.3%

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

   if 2e22 < m

   1. Initial program 80.3%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.2

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

   5. Simplified 3.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 2.8%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 36.9

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

   10. Simplified 36.9%

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

   11. Taylor expanded in k around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 36.4

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

   13. Simplified 36.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 0.0155:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

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 0.0155) a_m (* 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 <= 0.0155) {
		tmp = a_m;
	} 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 <= 0.0155d0) then
        tmp = a_m
    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 <= 0.0155) {
		tmp = a_m;
	} 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 <= 0.0155:
		tmp = a_m
	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 <= 0.0155)
		tmp = a_m;
	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 <= 0.0155)
		tmp = a_m;
	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, 0.0155], a$95$m, N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 0.0155

   1. Initial program 97.3%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 30.5%

      \[\leadsto \color{blue}{a} \]

   if 0.0155 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 3.4

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

   5. Simplified 3.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 11.6

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

   8. Simplified 11.6%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 23.1

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

   11. Simplified 23.1%

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

Alternative 16: 19.9% accurate, 134.0× speedup

Math

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

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))

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;
}

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
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;
}

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. pow-lowering-pow.f64 84.7

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

5. Simplified 84.7%

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

6. Taylor expanded in m around 0

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

8. Simplified 22.1%

   \[\leadsto \color{blue}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))