Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 97.7%

Time: 8.5s

Alternatives: 12

Speedup: 1.1×

• Could not converge on a ground truth

  1. a = 1.1030819434171483e-238
  2. k = -3.0240028693717434e-261
  3. m = 3.349827727720969e+92

• 21.3% 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: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 3.5) (/ t_0 (fma (+ k 10.0) k 1.0)) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 3.5) {
		tmp = t_0 / fma((k + 10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 3.5)
		tmp = Float64(t_0 / fma(Float64(k + 10.0), k, 1.0));
	else
		tmp = 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[m, 3.5], N[(t$95$0 / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 97.8

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if 3.5 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.5) (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.5) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.5)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.5], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.8

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 3.5 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.95 \cdot 10^{-12}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.46 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(-1 + \left(-1 + m\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.95e-12)
   (/ (* a (pow k m)) (fma 10.0 k 1.0))
   (if (<= m 1.46e-9)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* a (pow k (+ -1.0 (+ -1.0 m)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.95e-12) {
		tmp = (a * pow(k, m)) / fma(10.0, k, 1.0);
	} else if (m <= 1.46e-9) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = a * pow(k, (-1.0 + (-1.0 + m)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.95e-12)
		tmp = Float64(Float64(a * (k ^ m)) / fma(10.0, k, 1.0));
	elseif (m <= 1.46e-9)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(a * (k ^ Float64(-1.0 + Float64(-1.0 + m))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.95e-12], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.46e-9], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(-1.0 + N[(-1.0 + m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.95e-12

   1. Initial program 98.8%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.95e-12 < m < 1.4599999999999999e-9

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.5%

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

   if 1.4599999999999999e-9 < m

   1. Initial program 81.6%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 99.9%

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

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -128:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 1.46 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(-1 + \left(-1 + m\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -128.0)
   (* (pow k m) a)
   (if (<= m 1.46e-9)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* a (pow k (+ -1.0 (+ -1.0 m)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -128.0) {
		tmp = pow(k, m) * a;
	} else if (m <= 1.46e-9) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = a * pow(k, (-1.0 + (-1.0 + m)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -128.0)
		tmp = Float64((k ^ m) * a);
	elseif (m <= 1.46e-9)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(a * (k ^ Float64(-1.0 + Float64(-1.0 + m))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -128.0], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 1.46e-9], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(-1.0 + N[(-1.0 + m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -128

   1. Initial program 98.8%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -128 < m < 1.4599999999999999e-9

   1. Initial program 97.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

   if 1.4599999999999999e-9 < m

   1. Initial program 81.6%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 99.9%

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

Alternative 5: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -128 \lor \neg \left(m \leq 0.95\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -128.0) (not (<= m 0.95)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -128.0) || !(m <= 0.95)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -128.0) || !(m <= 0.95))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -128.0], N[Not[LessEqual[m, 0.95]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -128 or 0.94999999999999996 < m

   1. Initial program 90.3%

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

   3. Taylor expanded in k around inf

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. exp-prod N/A

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

      6. neg-mul-1 N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. rem-exp-log N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot \frac{{\color{blue}{k}}^{m}}{1} \]

      10. /-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -128 < m < 0.94999999999999996

   1. Initial program 97.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -128 \lor \neg \left(m \leq 0.95\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -128:\\ \;\;\;\;\frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -128.0)
   (/ (* (/ a (* k k)) 99.0) (* k k))
   (if (<= m 1.1) (/ a (fma (+ 10.0 k) k 1.0)) (* (- a) (* (* -99.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -128.0) {
		tmp = ((a / (k * k)) * 99.0) / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = -a * ((-99.0 * k) * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -128.0)
		tmp = Float64(Float64(Float64(a / Float64(k * k)) * 99.0) / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-a) * Float64(Float64(-99.0 * k) * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -128.0], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -128

   1. Initial program 98.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{99 \cdot \frac{a}{{k}^{2}}}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.0%

       \[\leadsto \frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k} \]

   if -128 < m < 1.1000000000000001

   1. Initial program 97.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   if 1.1000000000000001 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 29.7%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 62.8%

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

Alternative 7: 72.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -128:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -128.0)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma (+ 10.0 k) k 1.0)) (* (- a) (* (* -99.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -128.0) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = -a * ((-99.0 * k) * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -128.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-a) * Float64(Float64(-99.0 * k) * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -128.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -128

   1. Initial program 98.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 57.3%

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

   if -128 < m < 1.1000000000000001

   1. Initial program 97.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   if 1.1000000000000001 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 29.7%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 62.8%

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

Alternative 8: 62.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -128:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -128.0)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma 10.0 k 1.0)) (* (- a) (* (* -99.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -128.0) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = -a * ((-99.0 * k) * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -128.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(-a) * Float64(Float64(-99.0 * k) * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -128.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -128

   1. Initial program 98.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 57.3%

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

   if -128 < m < 1.1000000000000001

   1. Initial program 97.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.4%

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

   if 1.1000000000000001 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 29.7%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 62.8%

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

Alternative 9: 57.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.5e-133)
   (/ a (* k k))
   (if (<= m 0.95) (* (- a) -1.0) (* (- a) (* (* -99.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-133) {
		tmp = a / (k * k);
	} else if (m <= 0.95) {
		tmp = -a * -1.0;
	} else {
		tmp = -a * ((-99.0 * k) * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.5d-133)) then
        tmp = a / (k * k)
    else if (m <= 0.95d0) then
        tmp = -a * (-1.0d0)
    else
        tmp = -a * (((-99.0d0) * k) * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-133) {
		tmp = a / (k * k);
	} else if (m <= 0.95) {
		tmp = -a * -1.0;
	} else {
		tmp = -a * ((-99.0 * k) * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.5e-133:
		tmp = a / (k * k)
	elif m <= 0.95:
		tmp = -a * -1.0
	else:
		tmp = -a * ((-99.0 * k) * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.5e-133)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.95)
		tmp = Float64(Float64(-a) * -1.0);
	else
		tmp = Float64(Float64(-a) * Float64(Float64(-99.0 * k) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.5e-133)
		tmp = a / (k * k);
	elseif (m <= 0.95)
		tmp = -a * -1.0;
	else
		tmp = -a * ((-99.0 * k) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.5e-133], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[((-a) * -1.0), $MachinePrecision], N[((-a) * N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.5000000000000001e-133

   1. Initial program 98.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 13.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 56.1%

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

   if -1.5000000000000001e-133 < m < 0.94999999999999996

   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. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(-a\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 52.1%

       \[\leadsto \left(-a\right) \cdot -1 \]

   if 0.94999999999999996 < m

   1. Initial program 81.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 29.7%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 62.8%

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

Alternative 10: 46.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-300} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 7.4e-300) (not (<= k 0.1)))
   (/ a (* k k))
   (* (fma -10.0 k 1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 7.4e-300) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = fma(-10.0, k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 7.4e-300) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(fma(-10.0, k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 7.4e-300], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.4000000000000003e-300 or 0.10000000000000001 < k

   1. Initial program 89.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 6.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 48.1%

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

   if 7.4000000000000003e-300 < k < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 57.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-300} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;\left(-a\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7.2e+21) (* (- a) -1.0) (* (* a k) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.2e+21) {
		tmp = -a * -1.0;
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7.2d+21) then
        tmp = -a * (-1.0d0)
    else
        tmp = (a * k) * (-10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.2e+21) {
		tmp = -a * -1.0;
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 7.2e+21:
		tmp = -a * -1.0
	else:
		tmp = (a * k) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 7.2e+21)
		tmp = Float64(Float64(-a) * -1.0);
	else
		tmp = Float64(Float64(a * k) * -10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 7.2e+21)
		tmp = -a * -1.0;
	else
		tmp = (a * k) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 7.2e+21], N[((-a) * -1.0), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.2e21

   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. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(-a\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 29.1%

       \[\leadsto \left(-a\right) \cdot -1 \]

   if 7.2e21 < m

   1. Initial program 81.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 11.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 25.1%

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

Alternative 12: 20.7% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \left(-a\right) \cdot -1 \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* (- a) -1.0))

C

double code(double a, double k, double m) {
	return -a * -1.0;
}

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 * (-1.0d0)
end function

Java

public static double code(double a, double k, double m) {
	return -a * -1.0;
}

Python

def code(a, k, m):
	return -a * -1.0

Julia

function code(a, k, m)
	return Float64(Float64(-a) * -1.0)
end

MATLAB

function tmp = code(a, k, m)
	tmp = -a * -1.0;
end

Wolfram

code[a_, k_, m_] := N[((-a) * -1.0), $MachinePrecision]

Derivation

1. Initial program 93.0%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. metadata-eval N/A

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

   9. lft-mult-inverse N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. associate-+l+ N/A

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

   14. distribute-lft1-in N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*r* N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 50.3%

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

7. Applied rewrites 50.3%

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

8. Taylor expanded in k around 0

   \[\leadsto \left(-a\right) \cdot -1 \]
9. Step-by-step derivation

10. Applied rewrites 22.0%

    \[\leadsto \left(-a\right) \cdot -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))