Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.3%

Time: 12.8s

Alternatives: 16

Speedup: 1.2×

• 22.0% 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: 90.1% 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: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{\left(-m\right)}\\ \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (pow k (- m))))
   (if (<= k 0.00082) (/ a t_0) (/ (/ a k) (* k t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m);
	double tmp;
	if (k <= 0.00082) {
		tmp = a / t_0;
	} else {
		tmp = (a / k) / (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 ** -m
    if (k <= 0.00082d0) then
        tmp = a / t_0
    else
        tmp = (a / k) / (k * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, -m);
	double tmp;
	if (k <= 0.00082) {
		tmp = a / t_0;
	} else {
		tmp = (a / k) / (k * t_0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, -m)
	tmp = 0
	if k <= 0.00082:
		tmp = a / t_0
	else:
		tmp = (a / k) / (k * t_0)
	return tmp

Julia

function code(a, k, m)
	t_0 = k ^ Float64(-m)
	tmp = 0.0
	if (k <= 0.00082)
		tmp = Float64(a / t_0);
	else
		tmp = Float64(Float64(a / k) / Float64(k * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k ^ -m;
	tmp = 0.0;
	if (k <= 0.00082)
		tmp = a / t_0;
	else
		tmp = (a / k) / (k * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Power[k, (-m)], $MachinePrecision]}, If[LessEqual[k, 0.00082], N[(a / t$95$0), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / N[(k * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-4

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

      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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-neg.f64 93.5

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 8.1999999999999998e-4 < k

   1. Initial program 79.1%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-flip N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-neg.f64 94.2

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

   7. Applied rewrites 94.2%

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

Alternative 2: 38.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+294}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, -10, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ 1.0 (* k 10.0))))))
   (if (<= t_1 5e-29) t_0 (if (<= t_1 1e+294) (* a (fma k -10.0 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + (1.0 + (k * 10.0)));
	double tmp;
	if (t_1 <= 5e-29) {
		tmp = t_0;
	} else if (t_1 <= 1e+294) {
		tmp = a * fma(k, -10.0, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))))
	tmp = 0.0
	if (t_1 <= 5e-29)
		tmp = t_0;
	elseif (t_1 <= 1e+294)
		tmp = Float64(a * fma(k, -10.0, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-29], t$95$0, If[LessEqual[t$95$1, 1e+294], N[(a * N[(k * -10.0 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.99999999999999986e-29 or 1.00000000000000007e294 < (/.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 87.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 37.0

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

   8. Applied rewrites 37.0%

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

   if 4.99999999999999986e-29 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000007e294

   1. Initial program 99.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.2

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

   5. Applied rewrites 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 92.2

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.3

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

   10. Applied rewrites 86.3%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 10^{+294}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, -10, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot {k}^{m}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.00082) (/ a (pow k (- m))) (/ (* (/ a k) (pow k m)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / pow(k, -m);
	} else {
		tmp = ((a / k) * pow(k, m)) / 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 <= 0.00082d0) then
        tmp = a / (k ** -m)
    else
        tmp = ((a / k) * (k ** m)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = ((a / k) * Math.pow(k, m)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.00082:
		tmp = a / math.pow(k, -m)
	else:
		tmp = ((a / k) * math.pow(k, m)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.00082)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(Float64(Float64(a / k) * (k ^ m)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.00082)
		tmp = a / (k ^ -m);
	else
		tmp = ((a / k) * (k ^ m)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.00082], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / k), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-4

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

      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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-neg.f64 93.5

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 8.1999999999999998e-4 < k

   1. Initial program 79.1%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 94.2

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

   7. Applied rewrites 94.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot {k}^{m}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.00082) (/ a (pow k (- m))) (* (/ a k) (/ (pow k m) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / pow(k, -m);
	} else {
		tmp = (a / k) * (pow(k, m) / 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 <= 0.00082d0) then
        tmp = a / (k ** -m)
    else
        tmp = (a / k) * ((k ** m) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = (a / k) * (Math.pow(k, m) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.00082:
		tmp = a / math.pow(k, -m)
	else:
		tmp = (a / k) * (math.pow(k, m) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.00082)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.00082)
		tmp = a / (k ^ -m);
	else
		tmp = (a / k) * ((k ^ m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.00082], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-4

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

      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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-neg.f64 93.5

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 8.1999999999999998e-4 < k

   1. Initial program 79.1%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.2

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

   7. Applied rewrites 94.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.0) (* 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 <= 3.0) {
		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 <= 3.0)
		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, 3.0], 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 < 3

   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 96.3

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

   4. Applied rewrites 96.3%

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

   if 3 < m

   1. Initial program 73.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.6%

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

Alternative 6: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.00082) (/ a (pow k (- m))) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / pow(k, -m);
	} else {
		tmp = a * pow(k, (m + -2.0));
	}
	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 (k <= 0.00082d0) then
        tmp = a / (k ** -m)
    else
        tmp = a * (k ** (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.00082:
		tmp = a / math.pow(k, -m)
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.00082)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(a * (k ^ Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.00082)
		tmp = a / (k ^ -m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.00082], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-4

   1. Initial program 93.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

      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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-neg.f64 93.5

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 8.1999999999999998e-4 < k

   1. Initial program 79.1%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-div N/A

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

      10. lower-pow.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. metadata-eval 92.2

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

   7. Applied rewrites 92.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.49:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -1.75e-7)
     t_0
     (if (<= m 0.49) (* a (/ 1.0 (fma k (+ k 10.0) 1.0))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -1.75e-7) {
		tmp = t_0;
	} else if (m <= 0.49) {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -1.75e-7)
		tmp = t_0;
	elseif (m <= 0.49)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.75e-7], t$95$0, If[LessEqual[m, 0.49], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.74999999999999992e-7 or 0.48999999999999999 < m

   1. Initial program 86.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.74999999999999992e-7 < m < 0.48999999999999999

   1. Initial program 92.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 89.6

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

   7. Applied rewrites 89.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 0.49:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.00082) (* a (pow k m)) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m + -2.0));
	}
	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 (k <= 0.00082d0) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.00082) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.00082:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.00082)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.00082)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.00082], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-4

   1. Initial program 93.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 8.1999999999999998e-4 < k

   1. Initial program 79.1%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-div N/A

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

      10. lower-pow.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. metadata-eval 92.2

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

   7. Applied rewrites 92.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00082:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.2:\\ \;\;\;\;a \cdot \frac{1 - \frac{10 + \frac{-99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.2)
   (* a (/ (- 1.0 (/ (+ 10.0 (/ -99.0 k)) k)) (* k k)))
   (if (<= m 1.26)
     (* a (/ 1.0 (fma k (+ k 10.0) 1.0)))
     (* a (* k (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.2) {
		tmp = a * ((1.0 - ((10.0 + (-99.0 / k)) / k)) / (k * k));
	} else if (m <= 1.26) {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.2)
		tmp = Float64(a * Float64(Float64(1.0 - Float64(Float64(10.0 + Float64(-99.0 / k)) / k)) / Float64(k * k)));
	elseif (m <= 1.26)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.2], N[(a * N[(N[(1.0 - N[(N[(10.0 + N[(-99.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.26], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.2000000000000002

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 29.6

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

   5. Applied rewrites 29.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 29.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 29.6

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

   7. Applied rewrites 29.6%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{10 - 99 \cdot \frac{1}{k}}{k}}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{10 - 99 \cdot \frac{1}{k}}{k}}{{k}^{2}}} \cdot a \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{10 - 99 \cdot \frac{1}{k}}{k}\right)\right)}}{{k}^{2}} \cdot a \]

      3. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{1 - \frac{10 - 99 \cdot \frac{1}{k}}{k}}}{{k}^{2}} \cdot a \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \frac{10 - 99 \cdot \frac{1}{k}}{k}}}{{k}^{2}} \cdot a \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1 - \color{blue}{\frac{10 - 99 \cdot \frac{1}{k}}{k}}}{{k}^{2}} \cdot a \]

      6. sub-neg N/A

         \[\leadsto \frac{1 - \frac{\color{blue}{10 + \left(\mathsf{neg}\left(99 \cdot \frac{1}{k}\right)\right)}}{k}}{{k}^{2}} \cdot a \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1 - \frac{\color{blue}{10 + \left(\mathsf{neg}\left(99 \cdot \frac{1}{k}\right)\right)}}{k}}{{k}^{2}} \cdot a \]

      8. associate-*r/ N/A

         \[\leadsto \frac{1 - \frac{10 + \left(\mathsf{neg}\left(\color{blue}{\frac{99 \cdot 1}{k}}\right)\right)}{k}}{{k}^{2}} \cdot a \]

      9. metadata-eval N/A

         \[\leadsto \frac{1 - \frac{10 + \left(\mathsf{neg}\left(\frac{\color{blue}{99}}{k}\right)\right)}{k}}{{k}^{2}} \cdot a \]

      10. distribute-neg-frac N/A

          \[\leadsto \frac{1 - \frac{10 + \color{blue}{\frac{\mathsf{neg}\left(99\right)}{k}}}{k}}{{k}^{2}} \cdot a \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1 - \frac{10 + \color{blue}{\frac{\mathsf{neg}\left(99\right)}{k}}}{k}}{{k}^{2}} \cdot a \]

      12. metadata-eval N/A

          \[\leadsto \frac{1 - \frac{10 + \frac{\color{blue}{-99}}{k}}{k}}{{k}^{2}} \cdot a \]

      13. unpow2 N/A

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

      14. lower-*.f64 69.4

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

   10. Applied rewrites 69.4%

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

   if -2.2000000000000002 < m < 1.26000000000000001

   1. Initial program 92.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if 1.26000000000000001 < m

   1. Initial program 73.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 37.7

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a \cdot {k}^{4}} \]
   12. 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. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 76.8

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

   13. Applied rewrites 76.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.2:\\ \;\;\;\;a \cdot \frac{1 - \frac{10 + \frac{-99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.43)
   (/ a (* k k))
   (if (<= m 1.26)
     (* a (/ 1.0 (fma k (+ k 10.0) 1.0)))
     (* a (* k (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.43) {
		tmp = a / (k * k);
	} else if (m <= 1.26) {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.43)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.26)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.43], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.26], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.429999999999999993

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if -0.429999999999999993 < m < 1.26000000000000001

   1. Initial program 92.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 88.6

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

   5. Applied rewrites 88.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if 1.26000000000000001 < m

   1. Initial program 73.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 37.7

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a \cdot {k}^{4}} \]
   12. 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. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 76.8

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

   13. Applied rewrites 76.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.43)
   (/ a (* k k))
   (if (<= m 1.26) (/ a (fma k (+ k 10.0) 1.0)) (* a (* k (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.43) {
		tmp = a / (k * k);
	} else if (m <= 1.26) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.43)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.26)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.43], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.26], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.429999999999999993

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if -0.429999999999999993 < m < 1.26000000000000001

   1. Initial program 92.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if 1.26000000000000001 < m

   1. Initial program 73.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 37.7

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a \cdot {k}^{4}} \]
   12. 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. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 76.8

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

   13. Applied rewrites 76.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.5e-136)
   (/ a (* k k))
   (if (<= m 1.26) (/ a (fma k 10.0 1.0)) (* a (* k (* k (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.5e-136) {
		tmp = a / (k * k);
	} else if (m <= 1.26) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a * (k * (k * (k * k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.5e-136)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.26)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a * Float64(k * Float64(k * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.5e-136], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.26], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.50000000000000011e-136

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   if -6.50000000000000011e-136 < m < 1.26000000000000001

   1. Initial program 90.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if 1.26000000000000001 < m

   1. Initial program 73.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 37.7

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a \cdot {k}^{4}} \]
   12. 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. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 76.8

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

   13. Applied rewrites 76.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.26:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.05:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.5e-136)
   (/ a (* k k))
   (if (<= m 2.05)
     (/ a (fma k 10.0 1.0))
     (* a (fma k (fma k 99.0 -10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.5e-136) {
		tmp = a / (k * k);
	} else if (m <= 2.05) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.5e-136)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.05)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.5e-136], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.05], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.50000000000000011e-136

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   if -6.50000000000000011e-136 < m < 2.0499999999999998

   1. Initial program 90.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if 2.0499999999999998 < m

   1. Initial program 73.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 2.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 37.0

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

   10. Applied rewrites 37.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.05:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{-144}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.9e-144) (/ a (* k k)) (* a (fma k (fma k 99.0 -10.0) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.9e-144) {
		tmp = a / (k * k);
	} else {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.9e-144)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.9e-144], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.90000000000000015e-144

   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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 42.6

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 59.5

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3.90000000000000015e-144 < m

   1. Initial program 79.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 36.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + k\right)} + 1} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a} \]

      6. lower-/.f64 36.4

         \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot a \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \cdot a \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot a \]

      9. lower-+.f64 36.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot a \]

   7. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(k \cdot \left(99 \cdot k - 10\right) + 1\right)} \cdot a \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, 99 \cdot k - 10, 1\right)} \cdot a \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{99 \cdot k + \left(\mathsf{neg}\left(10\right)\right)}, 1\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{k \cdot 99} + \left(\mathsf{neg}\left(10\right)\right), 1\right) \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, k \cdot 99 + \color{blue}{-10}, 1\right) \cdot a \]

      6. lower-fma.f64 42.6

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(k, 99, -10\right)}, 1\right) \cdot a \]

   10. Applied rewrites 42.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.9 \cdot 10^{-144}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 21.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \mathsf{fma}\left(k, -10, 1\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* a (fma k -10.0 1.0)))

C

double code(double a, double k, double m) {
	return a * fma(k, -10.0, 1.0);
}

Julia

function code(a, k, m)
	return Float64(a * fma(k, -10.0, 1.0))
end

Wolfram

code[a_, k_, m_] := N[(a * N[(k * -10.0 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 39.1

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + k\right)} + 1} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   3. div-inv N/A

      \[\leadsto \color{blue}{a \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a} \]

   6. lower-/.f64 39.1

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot a \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \cdot a \]

   8. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot a \]

   9. lower-+.f64 39.1

      \[\leadsto \frac{1}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot a \]

7. Applied rewrites 39.1%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\left(1 + -10 \cdot k\right)} \cdot a \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right)} \cdot a \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{k \cdot -10} + 1\right) \cdot a \]

   3. lower-fma.f64 18.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -10, 1\right)} \cdot a \]

10. Applied rewrites 18.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(k, -10, 1\right)} \cdot a \]

11. Final simplification 18.8%

    \[\leadsto a \cdot \mathsf{fma}\left(k, -10, 1\right) \]
12. Add Preprocessing

Alternative 16: 20.8% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return 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 = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 88.6%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 39.1

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{a}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 17.4%

   \[\leadsto \frac{a}{\color{blue}{1}} \]
9. Step-by-step derivation

   1. /-rgt-identity 17.4

      \[\leadsto \color{blue}{a} \]

10. Applied rewrites 17.4%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))