Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 97.4%

Time: 11.8s

Alternatives: 15

Speedup: 1.0×

• 22.9% 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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 4e+155) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+155) {
		tmp = t_1;
	} else {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 4d+155) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+155) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 4e+155:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 4e+155)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 4e+155)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+155], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000003e155

   1. Initial program 97.2%

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

   if 4.00000000000000003e155 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 61.9%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. /-rgt-identity 100.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 4 \cdot 10^{+155}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 16.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (((a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= Double.POSITIVE_INFINITY) {
		tmp = k * (a * k);
	} else {
		tmp = k * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if ((a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= math.inf:
		tmp = k * (a * k)
	else:
		tmp = k * k
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (((a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k))) <= Inf)
		tmp = k * (a * k);
	else
		tmp = k * k;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   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

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 13.6

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

   5. Simplified 13.6%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 50.1

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

   8. Simplified 50.1%

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

4. Final simplification 17.0%

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

Alternative 3: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{t\_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 1.0)
     t_0
     (if (<= k 3.4e+152)
       (/ t_0 (* k k))
       (* a (/ (pow k m) (fma k 10.0 1.0)))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 1.0) {
		tmp = t_0;
	} else if (k <= 3.4e+152) {
		tmp = t_0 / (k * k);
	} else {
		tmp = a * (pow(k, m) / fma(k, 10.0, 1.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 1.0)
		tmp = t_0;
	elseif (k <= 3.4e+152)
		tmp = Float64(t_0 / Float64(k * k));
	else
		tmp = Float64(a * Float64((k ^ m) / fma(k, 10.0, 1.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.0], t$95$0, If[LessEqual[k, 3.4e+152], N[(t$95$0 / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1

   1. Initial program 94.8%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. /-rgt-identity 99.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 1 < k < 3.4000000000000002e152

   1. Initial program 99.9%

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

   3. Taylor expanded in k around -inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if 3.4000000000000002e152 < k

   1. Initial program 58.1%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 58.1

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 58.1

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

   4. Applied egg-rr 58.1%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.5

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

   7. Simplified 68.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.00125)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00125) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.00125)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.00125], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.00125000000000000003

   1. Initial program 96.3%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.3

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.8

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

   4. Applied egg-rr 96.8%

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

   if 0.00125000000000000003 < 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 k around 0

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

   5. Simplified 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. /-rgt-identity 100.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= 1.0d0) then
        tmp = t_0
    else
        tmp = t_0 / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (k <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_0 / (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 1.0:
		tmp = t_0
	else:
		tmp = t_0 / (k * k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(t_0 / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 1.0)
		tmp = t_0;
	else
		tmp = t_0 / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 94.8%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. /-rgt-identity 99.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

   7. Applied egg-rr 99.5%

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

   if 1 < k

   1. Initial program 79.2%

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

   3. Taylor expanded in k around -inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+69}:\\ \;\;\;\;\frac{k \cdot \left(a \cdot k\right)}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k \cdot k, 10\right), 1\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{k \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 9.2e+30)
     t_0
     (if (<= k 1.45e+69)
       (/ (* k (* a k)) (fma k (fma k (* k k) 10.0) 1.0))
       (if (<= k 1.15e+179) t_0 (/ -1.0 (* k (* k k))))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 9.2e+30) {
		tmp = t_0;
	} else if (k <= 1.45e+69) {
		tmp = (k * (a * k)) / fma(k, fma(k, (k * k), 10.0), 1.0);
	} else if (k <= 1.15e+179) {
		tmp = t_0;
	} else {
		tmp = -1.0 / (k * (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 9.2e+30)
		tmp = t_0;
	elseif (k <= 1.45e+69)
		tmp = Float64(Float64(k * Float64(a * k)) / fma(k, fma(k, Float64(k * k), 10.0), 1.0));
	elseif (k <= 1.15e+179)
		tmp = t_0;
	else
		tmp = Float64(-1.0 / Float64(k * Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9.2e+30], t$95$0, If[LessEqual[k, 1.45e+69], N[(N[(k * N[(a * k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision] + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+179], t$95$0, N[(-1.0 / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.2e30 or 1.4499999999999999e69 < k < 1.14999999999999997e179

   1. Initial program 93.4%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 91.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. /-rgt-identity 91.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.0

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

   7. Applied egg-rr 91.0%

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

   if 9.2e30 < k < 1.4499999999999999e69

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 6.9

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

   5. Simplified 6.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{k \cdot \left(k \cdot a\right)}{\color{blue}{1 + k \cdot \left(10 + {k}^{3}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{k \cdot \left(k \cdot a\right)}{\color{blue}{k \cdot \left(10 + {k}^{3}\right) + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{k \cdot \left(k \cdot a\right)}{\color{blue}{\mathsf{fma}\left(k, 10 + {k}^{3}, 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{k \cdot \left(k \cdot a\right)}{\mathsf{fma}\left(k, \color{blue}{{k}^{3} + 10}, 1\right)} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.6

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

   8. Simplified 91.6%

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

   if 1.14999999999999997e179 < k

   1. Initial program 62.0%

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

   3. Taylor expanded in m around inf

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

      1. metadata-eval N/A

         \[\leadsto \frac{a \cdot {k}^{\color{blue}{\left(3 + 1\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      2. pow-plus N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left({k}^{3} \cdot k\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot {k}^{3}\right) \cdot k}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{\frac{-1}{{k}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{k}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 62.4

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

   8. Simplified 62.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+69}:\\ \;\;\;\;\frac{k \cdot \left(a \cdot k\right)}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k \cdot k, 10\right), 1\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+179}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{k \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot \left(k \cdot k\right)\right)\\ \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;a \cdot \frac{99}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (* k (* k k)))))
   (if (<= m 0.00125) (* a (/ 99.0 t_0)) (* a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k * (k * k));
	double tmp;
	if (m <= 0.00125) {
		tmp = a * (99.0 / t_0);
	} else {
		tmp = a * t_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) :: t_0
    real(8) :: tmp
    t_0 = k * (k * (k * k))
    if (m <= 0.00125d0) then
        tmp = a * (99.0d0 / t_0)
    else
        tmp = a * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k * (k * k));
	double tmp;
	if (m <= 0.00125) {
		tmp = a * (99.0 / t_0);
	} else {
		tmp = a * t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k * (k * k))
	tmp = 0
	if m <= 0.00125:
		tmp = a * (99.0 / t_0)
	else:
		tmp = a * t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k * Float64(k * k)))
	tmp = 0.0
	if (m <= 0.00125)
		tmp = Float64(a * Float64(99.0 / t_0));
	else
		tmp = Float64(a * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k * (k * k));
	tmp = 0.0;
	if (m <= 0.00125)
		tmp = a * (99.0 / t_0);
	else
		tmp = a * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 0.00125], N[(a * N[(99.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 0.00125000000000000003

   1. Initial program 96.3%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.3

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.8

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in k around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \frac{99 \cdot \frac{1}{k} - 10}{k} - 1}{\color{blue}{-1 \cdot k}} \cdot a \]

      4. *-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{99 \cdot \frac{1}{k} - 10}{k} - 1}{\color{blue}{k \cdot -1}} \cdot a \]

   7. Simplified 26.2%

      \[\leadsto \color{blue}{\frac{1 + \frac{-10 + \frac{99}{k}}{k}}{k}} \cdot a \]

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{99}{{k}^{\color{blue}{\left(3 + 1\right)}}} \cdot a \]

      3. pow-plus N/A

         \[\leadsto \frac{99}{\color{blue}{{k}^{3} \cdot k}} \cdot a \]

      4. *-commutative N/A

         \[\leadsto \frac{99}{\color{blue}{k \cdot {k}^{3}}} \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{99}{\color{blue}{k \cdot {k}^{3}}} \cdot a \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.4

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

   10. Simplified 53.4%

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

   if 0.00125000000000000003 < 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 k around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot {k}^{\color{blue}{\left(3 + 1\right)}} \]

      3. pow-plus N/A

         \[\leadsto a \cdot \color{blue}{\left({k}^{3} \cdot k\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.5

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

   8. Simplified 82.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;a \cdot \frac{99}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;\frac{a \cdot 99}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.00125) (/ (* a 99.0) (* k k)) (* a (* k (* k (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00125) {
		tmp = (a * 99.0) / (k * k);
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.00125d0) then
        tmp = (a * 99.0d0) / (k * k)
    else
        tmp = a * (k * (k * (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00125) {
		tmp = (a * 99.0) / (k * k);
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.00125:
		tmp = (a * 99.0) / (k * k)
	else:
		tmp = a * (k * (k * (k * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.00125)
		tmp = Float64(Float64(a * 99.0) / Float64(k * k));
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.00125)
		tmp = (a * 99.0) / (k * k);
	else
		tmp = a * (k * (k * (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.00125], N[(N[(a * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.00125000000000000003

   1. Initial program 96.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied egg-rr 68.9%

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

   5. Taylor expanded in k around -inf

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

   7. Simplified 14.2%

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

   8. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.0

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

   10. Simplified 48.0%

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

   if 0.00125000000000000003 < 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 k around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot {k}^{\color{blue}{\left(3 + 1\right)}} \]

      3. pow-plus N/A

         \[\leadsto a \cdot \color{blue}{\left({k}^{3} \cdot k\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.5

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

   8. Simplified 82.5%

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

Alternative 9: 52.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot k\right)\\ \mathbf{if}\;m \leq 1.22 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (* k k))))
   (if (<= m 1.22e-45) (/ -1.0 t_0) (* a (* k t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k * k);
	double tmp;
	if (m <= 1.22e-45) {
		tmp = -1.0 / t_0;
	} else {
		tmp = a * (k * t_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) :: t_0
    real(8) :: tmp
    t_0 = k * (k * k)
    if (m <= 1.22d-45) then
        tmp = (-1.0d0) / t_0
    else
        tmp = a * (k * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k * k);
	double tmp;
	if (m <= 1.22e-45) {
		tmp = -1.0 / t_0;
	} else {
		tmp = a * (k * t_0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k * k)
	tmp = 0
	if m <= 1.22e-45:
		tmp = -1.0 / t_0
	else:
		tmp = a * (k * t_0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k * k))
	tmp = 0.0
	if (m <= 1.22e-45)
		tmp = Float64(-1.0 / t_0);
	else
		tmp = Float64(a * Float64(k * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k * k);
	tmp = 0.0;
	if (m <= 1.22e-45)
		tmp = -1.0 / t_0;
	else
		tmp = a * (k * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 1.22e-45], N[(-1.0 / t$95$0), $MachinePrecision], N[(a * N[(k * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 1.22000000000000007e-45

   1. Initial program 96.7%

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

   3. Taylor expanded in m around inf

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

      1. metadata-eval N/A

         \[\leadsto \frac{a \cdot {k}^{\color{blue}{\left(3 + 1\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      2. pow-plus N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left({k}^{3} \cdot k\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot {k}^{3}\right) \cdot k}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 2.3

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

   5. Simplified 2.3%

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

   6. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{\frac{-1}{{k}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{k}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 41.6

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

   8. Simplified 41.6%

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

   if 1.22000000000000007e-45 < m

   1. Initial program 72.5%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 98.9%

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

   6. Taylor expanded in m around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot {k}^{\color{blue}{\left(3 + 1\right)}} \]

      3. pow-plus N/A

         \[\leadsto a \cdot \color{blue}{\left({k}^{3} \cdot k\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.0

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

   8. Simplified 78.0%

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

Alternative 10: 40.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.35) (/ a k) (* a (* k (* k (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / k;
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.35d0) then
        tmp = a / k
    else
        tmp = a * (k * (k * (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / k;
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.35:
		tmp = a / k
	else:
		tmp = a * (k * (k * (k * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.35)
		tmp = Float64(a / k);
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.35)
		tmp = a / k;
	else
		tmp = a * (k * (k * (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.35], N[(a / k), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3500000000000001

   1. Initial program 96.3%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.3

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in k around -inf

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

      1. lower-/.f64 21.5

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

   7. Simplified 21.5%

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

   if 1.3500000000000001 < m

   1. Initial program 71.6%

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

   3. Taylor expanded in k around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot {k}^{\color{blue}{\left(3 + 1\right)}} \]

      3. pow-plus N/A

         \[\leadsto a \cdot \color{blue}{\left({k}^{3} \cdot k\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(k \cdot {k}^{3}\right)} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 83.5

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

   8. Simplified 83.5%

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

Alternative 11: 33.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.35) (/ a k) (* a (* k k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / k;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.35d0) then
        tmp = a / k
    else
        tmp = a * (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / k;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.35:
		tmp = a / k
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.35)
		tmp = Float64(a / k);
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.35)
		tmp = a / k;
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.35], N[(a / k), $MachinePrecision], N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3500000000000001

   1. Initial program 96.3%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.3

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in k around -inf

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

      1. lower-/.f64 21.5

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

   7. Simplified 21.5%

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

   if 1.3500000000000001 < m

   1. Initial program 71.6%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.8

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

   5. Simplified 51.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 65.6

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

   7. Applied egg-rr 65.6%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.5% accurate, 12.2× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	return a * (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 * k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 18.4

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

5. Simplified 18.4%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 22.8

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

7. Applied egg-rr 22.8%

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

8. Final simplification 22.8%

   \[\leadsto a \cdot \left(k \cdot k\right) \]
9. Add Preprocessing

Alternative 13: 14.2% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ k \cdot k \end{array} \]

FPCore

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

C

double code(double a, double k, double m) {
	return 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 = k * k
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 18.4

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

5. Simplified 18.4%

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

6. Taylor expanded in k around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 13.7

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

8. Simplified 13.7%

   \[\leadsto \color{blue}{k \cdot k} \]
9. Add Preprocessing

Alternative 14: 9.4% accurate, 22.3× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	return a * 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

   1. lift-pow.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 88.5

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate-+l+ N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-+.f64 88.9

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

4. Applied egg-rr 88.9%

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

5. Taylor expanded in k around 0

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

   1. lower-*.f64 9.9

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

7. Simplified 9.9%

   \[\leadsto \color{blue}{a \cdot k} \]
8. Add Preprocessing

Alternative 15: 2.9% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := 1.0

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 15.9

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

5. Simplified 15.9%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 2.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))