Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.0%

Time: 9.9s

Alternatives: 12

Speedup: 1.1×

• Could not converge on a ground truth

  1. a = 3.3368725421882266e-105
  2. k = -3.889953689184385e+207
  3. m = -7.94700797932475e+235

• 21.9% 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: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = pow(k, m) * a;
	} else {
		tmp = (pow(k, (-1.0 + m)) * a) / 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 (k <= 1.0d0) then
        tmp = (k ** m) * a
    else
        tmp = ((k ** ((-1.0d0) + m)) * a) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = (Math.pow(k, (-1.0 + m)) * a) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = math.pow(k, m) * a
	else:
		tmp = (math.pow(k, (-1.0 + m)) * a) / k
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = (k ^ m) * a;
	else
		tmp = ((k ^ (-1.0 + m)) * a) / k;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.9%

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

   3. Taylor expanded in k around 0

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

      1. *-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1 < k

   1. Initial program 81.4%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. exp-prod N/A

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

      7. neg-mul-1 N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. rem-exp-log 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 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 98.9%

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

Alternative 2: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -29000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -29000000.0)
     t_0
     (if (<= m 2.55e-25) (* (/ 1.0 (fma (+ 10.0 k) k 1.0)) a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -29000000.0) {
		tmp = t_0;
	} else if (m <= 2.55e-25) {
		tmp = (1.0 / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -29000000.0)
		tmp = t_0;
	elseif (m <= 2.55e-25)
		tmp = Float64(Float64(1.0 / fma(Float64(10.0 + k), k, 1.0)) * a);
	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, -29000000.0], t$95$0, If[LessEqual[m, 2.55e-25], N[(N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.9e7 or 2.5500000000000001e-25 < m

   1. Initial program 85.5%

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

   3. Taylor expanded in k around 0

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

      1. *-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

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < m < 2.5500000000000001e-25

   1. Initial program 97.6%

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

      1. 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 96.5

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

      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 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 95.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -29000000:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 2.55 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* (pow k m) a) (* (pow k (+ (+ -1.0 m) -1.0)) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = pow(k, m) * a;
	} else {
		tmp = pow(k, ((-1.0 + m) + -1.0)) * a;
	}
	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 (k <= 1.0d0) then
        tmp = (k ** m) * a
    else
        tmp = (k ** (((-1.0d0) + m) + (-1.0d0))) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = Math.pow(k, ((-1.0 + m) + -1.0)) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = math.pow(k, m) * a
	else:
		tmp = math.pow(k, ((-1.0 + m) + -1.0)) * a
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.9%

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

   3. Taylor expanded in k around 0

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

      1. *-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1 < k

   1. Initial program 81.4%

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

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. exp-prod N/A

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

      7. neg-mul-1 N/A

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. rem-exp-log 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 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 96.0%

      \[\leadsto a \cdot \color{blue}{{k}^{\left(-1 + \left(-1 + m\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

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

Alternative 4: 76.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.6:\\ \;\;\;\;\frac{99 \cdot \frac{\frac{a}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.6)
   (/ (* 99.0 (/ (/ a k) k)) (* k k))
   (if (<= m 0.92)
     (* (/ 1.0 (fma (+ 10.0 k) k 1.0)) a)
     (* (* (* 99.0 k) k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.6) {
		tmp = (99.0 * ((a / k) / k)) / (k * k);
	} else if (m <= 0.92) {
		tmp = (1.0 / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.6)
		tmp = Float64(Float64(99.0 * Float64(Float64(a / k) / k)) / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(Float64(1.0 / fma(Float64(10.0 + k), k, 1.0)) * a);
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.6], N[(N[(99.0 * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 34.8%

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

   7. Applied rewrites 22.9%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 68.1%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 73.2%

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

   if -0.599999999999999978 < m < 0.92000000000000004

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. 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 97.7

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 95.4%

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

   if 0.92000000000000004 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.6:\\ \;\;\;\;\frac{99 \cdot \frac{\frac{a}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -29000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -29000000.0)
   (/ a (* k k))
   (if (<= m 0.92)
     (* (/ 1.0 (fma (+ 10.0 k) k 1.0)) a)
     (* (* (* 99.0 k) k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -29000000.0) {
		tmp = a / (k * k);
	} else if (m <= 0.92) {
		tmp = (1.0 / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -29000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(Float64(1.0 / fma(Float64(10.0 + k), k, 1.0)) * a);
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -29000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.9e7

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 34.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.6%

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

   if -2.9e7 < m < 0.92000000000000004

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. 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 96.7

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

      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 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 94.4%

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

   if 0.92000000000000004 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -29000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -29000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -29000000.0)
   (/ a (* k k))
   (if (<= m 0.92) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* 99.0 k) k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -29000000.0) {
		tmp = a / (k * k);
	} else if (m <= 0.92) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -29000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -29000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.9e7

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. 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. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 34.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 62.6%

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

   if -2.9e7 < m < 0.92000000000000004

   1. Initial program 97.7%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 94.4%

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

   if 0.92000000000000004 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

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

Alternative 7: 62.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.5e-41)
   (/ a (* k k))
   (if (<= m 1.2) (/ a (fma 10.0 k 1.0)) (* (* (* 99.0 k) k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-41) {
		tmp = a / (k * k);
	} else if (m <= 1.2) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.5e-41)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.2)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.5e-41], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.2], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.49999999999999994e-41

   1. Initial program 98.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 35.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.4%

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

   if -1.49999999999999994e-41 < m < 1.19999999999999996

   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. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 97.8%

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

   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 72.6%

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

   if 1.19999999999999996 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

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

Alternative 8: 57.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.205:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.8e-74)
   (/ a (* k k))
   (if (<= m 0.205) (* 1.0 a) (* (* (* 99.0 k) k) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.8e-74) {
		tmp = a / (k * k);
	} else if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	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 <= (-1.8d-74)) then
        tmp = a / (k * k)
    else if (m <= 0.205d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((99.0d0 * k) * k) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.8e-74) {
		tmp = a / (k * k);
	} else if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.8e-74:
		tmp = a / (k * k)
	elif m <= 0.205:
		tmp = 1.0 * a
	else:
		tmp = ((99.0 * k) * k) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.8e-74)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.205)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.8e-74)
		tmp = a / (k * k);
	elseif (m <= 0.205)
		tmp = 1.0 * a;
	else
		tmp = ((99.0 * k) * k) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.8e-74], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.205], N[(1.0 * a), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8000000000000001e-74

   1. Initial program 99.0%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 38.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 61.1%

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

   if -1.8000000000000001e-74 < m < 0.204999999999999988

   1. Initial program 98.7%

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

   3. Taylor expanded in k around 0

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

      1. *-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 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto 1 \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 59.7%

      \[\leadsto 1 \cdot a \]

   if 0.204999999999999988 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

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

Alternative 9: 40.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.205:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(99 \cdot k\right) \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.205) (* 1.0 a) (* (* (* 99.0 k) k) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	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 <= 0.205d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((99.0d0 * k) * k) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((99.0 * k) * k) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.205:
		tmp = 1.0 * a
	else:
		tmp = ((99.0 * k) * k) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.205)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(99.0 * k) * k) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.205)
		tmp = 1.0 * a;
	else
		tmp = ((99.0 * k) * k) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.205], N[(1.0 * a), $MachinePrecision], N[(N[(N[(99.0 * k), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.204999999999999988

   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

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. 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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in m around 0

      \[\leadsto 1 \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 30.5%

      \[\leadsto 1 \cdot a \]

   if 0.204999999999999988 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   7. Applied rewrites 2.6%

      \[\leadsto \frac{a}{\mathsf{fma}\left(\mathsf{fma}\left(100, k \cdot k, -1\right), \color{blue}{\frac{1}{\mathsf{fma}\left(10, k, -1\right)}}, k \cdot k\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 40.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 67.5%

       \[\leadsto \left(\left(99 \cdot k\right) \cdot k\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.205:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.205) (* 1.0 a) (* (* (* a k) k) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	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 <= 0.205d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((a * k) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.205) {
		tmp = 1.0 * a;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.205:
		tmp = 1.0 * a
	else:
		tmp = ((a * k) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.205)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.205)
		tmp = 1.0 * a;
	else
		tmp = ((a * k) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.205], N[(1.0 * a), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.204999999999999988

   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

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. 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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in m around 0

      \[\leadsto 1 \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 30.5%

      \[\leadsto 1 \cdot a \]

   if 0.204999999999999988 < m

   1. Initial program 70.9%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 35.1%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.5%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 99 \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.205:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.6e+15) (* 1.0 a) (* (* a k) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.6e+15) {
		tmp = 1.0 * a;
	} 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 <= 3.6d+15) then
        tmp = 1.0d0 * a
    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 <= 3.6e+15) {
		tmp = 1.0 * a;
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 3.6e+15:
		tmp = 1.0 * a
	else:
		tmp = (a * k) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.6e+15)
		tmp = Float64(1.0 * a);
	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 <= 3.6e+15)
		tmp = 1.0 * a;
	else
		tmp = (a * k) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.6e+15], N[(1.0 * a), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.6e15

   1. Initial program 97.7%

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

   3. Taylor expanded in k around 0

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

      1. *-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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto 1 \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 30.0%

      \[\leadsto 1 \cdot a \]

   if 3.6e15 < m

   1. Initial program 72.0%

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

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. lft-mult-inverse N/A

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

      11. distribute-rgt-in N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. associate-+r+ N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lft-mult-inverse N/A

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

      19. 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 2.8%

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

   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 10.0%

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

   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 24.1%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 21.4% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot a \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.5%

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

3. Taylor expanded in k around 0

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

   1. *-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 85.6

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

5. Applied rewrites 85.6%

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

6. Taylor expanded in m around 0

   \[\leadsto 1 \cdot a \]
7. Step-by-step derivation

8. Applied rewrites 21.6%

   \[\leadsto 1 \cdot a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))