Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 99.7%

Time: 12.6s

Alternatives: 14

Speedup: 1.1×

• 21.8% 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: 90.5% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \left(k \cdot \frac{{\left(\frac{1}{k}\right)}^{m}}{a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 1e-160)
     t_0
     (if (<= k 3e+23)
       (/ 1.0 (/ (fma k (+ k 10.0) 1.0) t_0))
       (/ 1.0 (* k (* k (/ (pow (/ 1.0 k) m) a))))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 1e-160) {
		tmp = t_0;
	} else if (k <= 3e+23) {
		tmp = 1.0 / (fma(k, (k + 10.0), 1.0) / t_0);
	} else {
		tmp = 1.0 / (k * (k * (pow((1.0 / k), m) / a)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 1e-160)
		tmp = t_0;
	elseif (k <= 3e+23)
		tmp = Float64(1.0 / Float64(fma(k, Float64(k + 10.0), 1.0) / t_0));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k * Float64((Float64(1.0 / k) ^ m) / a))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1e-160], t$95$0, If[LessEqual[k, 3e+23], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k * N[(N[Power[N[(1.0 / k), $MachinePrecision], m], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.9999999999999999e-161

   1. Initial program 93.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 9.9999999999999999e-161 < k < 3.0000000000000001e23

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   if 3.0000000000000001e23 < k

   1. Initial program 80.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.7

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in k around inf

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

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{k \cdot \color{blue}{\left(k \cdot \frac{{\left(\frac{1}{k}\right)}^{m}}{a}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-160}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a \cdot {k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \left(k \cdot \frac{{\left(\frac{1}{k}\right)}^{m}}{a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{k}^{\left(-m\right)}}{a}\\ \mathbf{if}\;k \leq 10^{-160}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_0 \cdot \left(k + 10\right), k, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (pow k (- m)) a)))
   (if (<= k 1e-160) (* a (pow k m)) (/ 1.0 (fma (* t_0 (+ k 10.0)) k t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m) / a;
	double tmp;
	if (k <= 1e-160) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / fma((t_0 * (k + 10.0)), k, t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ Float64(-m)) / a)
	tmp = 0.0
	if (k <= 1e-160)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / fma(Float64(t_0 * Float64(k + 10.0)), k, t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[k, 1e-160], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * k + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999999e-161

   1. Initial program 93.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 9.9999999999999999e-161 < k

   1. Initial program 88.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 88.0

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

   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 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))