Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 99.7%

Time: 12.3s

Alternatives: 17

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: 90.6% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k}{t\_0} + \frac{10}{t\_0}, \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 2e-36)
     t_0
     (/ 1.0 (fma k (+ (/ k t_0) (/ 10.0 t_0)) (/ 1.0 t_0))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 2e-36) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(k, ((k / t_0) + (10.0 / t_0)), (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 2e-36)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(k, Float64(Float64(k / t_0) + Float64(10.0 / t_0)), Float64(1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-36], t$95$0, N[(1.0 / N[(k * N[(N[(k / t$95$0), $MachinePrecision] + N[(10.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e-36

   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 k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.9999999999999999e-36 < k

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.4

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 2: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), k, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+268)
     (/ 1.0 (fma (* (/ (pow k (- m)) a) (+ k 10.0)) k (/ 1.0 a)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+268) {
		tmp = 1.0 / fma(((pow(k, -m) / a) * (k + 10.0)), k, (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+268)
		tmp = Float64(1.0 / fma(Float64(Float64((k ^ Float64(-m)) / a) * Float64(k + 10.0)), k, Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+268], N[(1.0 / N[(N[(N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.9999999999999999e268

   1. Initial program 97.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.3

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 78.3

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

   7. Applied rewrites 78.3%

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

   9. Applied rewrites 85.4%

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

   10. Taylor expanded in m around 0

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

   12. Applied rewrites 85.6%

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

   if 1.9999999999999999e268 < (/.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 61.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{{k}^{\left(-m\right)}}{a} \cdot \left(k + 10\right), k, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))