Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 98.9%

Time: 13.5s

Alternatives: 16

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 3.7041061149655524e-29
  2. k = -2.1400279247121017e-203
  3. m = 1.2485289832697376e+220

• 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: 90.2% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.6e-11) (* (pow k m) a) (/ (* a (/ (pow k m) k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-11) {
		tmp = pow(k, m) * a;
	} else {
		tmp = (a * (pow(k, m) / k)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.6d-11) then
        tmp = (k ** m) * a
    else
        tmp = (a * ((k ** m) / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-11) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = (a * (Math.pow(k, m) / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.6e-11:
		tmp = math.pow(k, m) * a
	else:
		tmp = (a * (math.pow(k, m) / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.6e-11)
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.6e-11)
		tmp = (k ^ m) * a;
	else
		tmp = (a * ((k ^ m) / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.6e-11], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999997e-11

   1. Initial program 92.0%

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

   3. Taylor expanded in k around 0

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

      1. rem-exp-log N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. exp-prod N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} + 0} \]

      9. *-commutative N/A

         \[\leadsto a \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + 0 \]

      10. associate-*r* N/A

          \[\leadsto a \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} + 0 \]

      11. exp-prod N/A

          \[\leadsto a \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} + 0 \]

      12. neg-mul-1 N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} + 0 \]

      13. log-rec N/A

          \[\leadsto a \cdot {\left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log k\right)\right)}\right)}\right)}^{m} + 0 \]

      14. remove-double-neg N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} + 0 \]

      15. rem-exp-log N/A

          \[\leadsto a \cdot {\color{blue}{k}}^{m} + 0 \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, {k}^{m}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. pow-lowering-pow.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.59999999999999997e-11 < k

   1. Initial program 82.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. pow-lowering-pow.f64 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.6e-11) (* (pow k m) a) (/ (* a (pow k (+ m -1.0))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-11) {
		tmp = pow(k, m) * a;
	} else {
		tmp = (a * pow(k, (m + -1.0))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.6d-11) then
        tmp = (k ** m) * a
    else
        tmp = (a * (k ** (m + (-1.0d0)))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-11) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = (a * Math.pow(k, (m + -1.0))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.6e-11:
		tmp = math.pow(k, m) * a
	else:
		tmp = (a * math.pow(k, (m + -1.0))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.6e-11)
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(Float64(a * (k ^ Float64(m + -1.0))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.6e-11)
		tmp = (k ^ m) * a;
	else
		tmp = (a * (k ^ (m + -1.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.6e-11], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[(a * N[Power[k, N[(m + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999997e-11

   1. Initial program 92.0%

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

   3. Taylor expanded in k around 0

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

      1. rem-exp-log N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. exp-prod N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} + 0} \]

      9. *-commutative N/A

         \[\leadsto a \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + 0 \]

      10. associate-*r* N/A

          \[\leadsto a \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} + 0 \]

      11. exp-prod N/A

          \[\leadsto a \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} + 0 \]

      12. neg-mul-1 N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} + 0 \]

      13. log-rec N/A

          \[\leadsto a \cdot {\left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log k\right)\right)}\right)}\right)}^{m} + 0 \]

      14. remove-double-neg N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} + 0 \]

      15. rem-exp-log N/A

          \[\leadsto a \cdot {\color{blue}{k}}^{m} + 0 \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, {k}^{m}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. pow-lowering-pow.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.59999999999999997e-11 < k

   1. Initial program 82.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. pow-lowering-pow.f64 98.8

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

   7. Applied egg-rr 98.8%

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. div-inv N/A

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

      4. inv-pow N/A

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

      5. pow-prod-up N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. +-lowering-+.f64 98.6

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -2.75 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.98:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -2.75e-17)
     t_0
     (if (<= m 0.98) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -2.75e-17) {
		tmp = t_0;
	} else if (m <= 0.98) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -2.75e-17)
		tmp = t_0;
	elseif (m <= 0.98)
		tmp = Float64(a / 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[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -2.75e-17], t$95$0, If[LessEqual[m, 0.98], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.75e-17 or 0.97999999999999998 < m

   1. Initial program 84.7%

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

   3. Taylor expanded in k around 0

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

      1. rem-exp-log N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. exp-prod N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} + 0} \]

      9. *-commutative N/A

         \[\leadsto a \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + 0 \]

      10. associate-*r* N/A

          \[\leadsto a \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} + 0 \]

      11. exp-prod N/A

          \[\leadsto a \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} + 0 \]

      12. neg-mul-1 N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} + 0 \]

      13. log-rec N/A

          \[\leadsto a \cdot {\left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log k\right)\right)}\right)}\right)}^{m} + 0 \]

      14. remove-double-neg N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} + 0 \]

      15. rem-exp-log N/A

          \[\leadsto a \cdot {\color{blue}{k}}^{m} + 0 \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, {k}^{m}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. pow-lowering-pow.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   if -2.75e-17 < m < 0.97999999999999998

   1. Initial program 93.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.75 \cdot 10^{-17}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 0.98:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.6e-11) (* (pow k m) a) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-11) {
		tmp = pow(k, m) * a;
	} 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 <= 1.6d-11) then
        tmp = (k ** m) * a
    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 <= 1.6e-11) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.6e-11:
		tmp = math.pow(k, m) * a
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.6e-11)
		tmp = Float64((k ^ m) * a);
	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 <= 1.6e-11)
		tmp = (k ^ m) * a;
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.6e-11], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999997e-11

   1. Initial program 92.0%

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

   3. Taylor expanded in k around 0

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

      1. rem-exp-log N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. exp-prod N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)} + 0} \]

      9. *-commutative N/A

         \[\leadsto a \cdot e^{-1 \cdot \color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + 0 \]

      10. associate-*r* N/A

          \[\leadsto a \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}} + 0 \]

      11. exp-prod N/A

          \[\leadsto a \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}} + 0 \]

      12. neg-mul-1 N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} + 0 \]

      13. log-rec N/A

          \[\leadsto a \cdot {\left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log k\right)\right)}\right)}\right)}^{m} + 0 \]

      14. remove-double-neg N/A

          \[\leadsto a \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m} + 0 \]

      15. rem-exp-log N/A

          \[\leadsto a \cdot {\color{blue}{k}}^{m} + 0 \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, {k}^{m}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. pow-lowering-pow.f64 99.1

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

   7. Applied egg-rr 99.1%

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

   if 1.59999999999999997e-11 < k

   1. Initial program 82.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow-div N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval 92.8

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

   7. Applied egg-rr 92.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(9801, \frac{a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}, \mathsf{fma}\left(\frac{a}{k \cdot k}, 99, a\right)\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (/
    (fma 9801.0 (/ a (* (* k k) (* k k))) (fma (/ a (* k k)) 99.0 a))
    (* k k))
   (if (<= m 1.4) (/ a (fma k (+ k 10.0) 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = fma(9801.0, (a / ((k * k) * (k * k))), fma((a / (k * k)), 99.0, a)) / (k * k);
	} else if (m <= 1.4) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(fma(9801.0, Float64(a / Float64(Float64(k * k) * Float64(k * k))), fma(Float64(a / Float64(k * k)), 99.0, a)) / Float64(k * k));
	elseif (m <= 1.4)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], N[(N[(9801.0 * N[(a / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0 + a), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.4], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 41.6

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

   7. Applied egg-rr 41.6%

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

   8. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 40.3

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

   10. Simplified 40.3%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\left(a + 9801 \cdot \frac{a}{{k}^{4}}\right) - -99 \cdot \frac{a}{{k}^{2}}}{{k}^{2}}} \]

   13. Simplified 69.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9801, \frac{a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}, \mathsf{fma}\left(\frac{a}{k \cdot k}, 99, a\right)\right)}{k \cdot k}} \]

   if -1.52 < m < 1.3999999999999999

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1.3999999999999999 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(9801, \frac{a}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}, \mathsf{fma}\left(\frac{a}{k \cdot k}, 99, a\right)\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;a \cdot \frac{1 - \frac{10 + \frac{-99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (* a (/ (- 1.0 (/ (+ 10.0 (/ -99.0 k)) k)) (* k k)))
   (if (<= m 1.35) (/ a (fma k (+ k 10.0) 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = a * ((1.0 - ((10.0 + (-99.0 / k)) / k)) / (k * k));
	} else if (m <= 1.35) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(a * Float64(Float64(1.0 - Float64(Float64(10.0 + Float64(-99.0 / k)) / k)) / Float64(k * k)));
	elseif (m <= 1.35)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], 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, 1.35], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 37.8

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

   7. Applied egg-rr 37.8%

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

   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. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.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. +-lowering-+.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. metadata-eval N/A

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

      12. /-lowering-/.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.9

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

   10. Simplified 67.9%

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

   if -1.52 < m < 1.3500000000000001

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1.3500000000000001 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;a \cdot \frac{1 - \frac{10 + \frac{-99}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{99}{k} - 10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (/ (fma (/ a k) (- (/ 99.0 k) 10.0) a) (* k k))
   (if (<= m 1.0) (/ a (fma k (+ k 10.0) 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = fma((a / k), ((99.0 / k) - 10.0), a) / (k * k);
	} else if (m <= 1.0) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(fma(Float64(a / k), Float64(Float64(99.0 / k) - 10.0), a) / Float64(k * k));
	elseif (m <= 1.0)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], N[(N[(N[(a / k), $MachinePrecision] * N[(N[(99.0 / k), $MachinePrecision] - 10.0), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.0], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 37.8

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

   7. Applied egg-rr 37.8%

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

   8. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]

   10. Simplified 65.5%

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

   if -1.52 < m < 1

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{99}{k} - 10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 99, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (/ (fma (/ a (* k k)) 99.0 a) (* k k))
   (if (<= m 1.25) (/ a (fma k (+ k 10.0) 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = fma((a / (k * k)), 99.0, a) / (k * k);
	} else if (m <= 1.25) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(fma(Float64(a / Float64(k * k)), 99.0, a) / Float64(k * k));
	elseif (m <= 1.25)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0 + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.25], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 41.6

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

   7. Applied egg-rr 41.6%

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

   8. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 40.3

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

   10. Simplified 40.3%

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

   11. Taylor expanded in k around inf

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

       1. metadata-eval N/A

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

       2. associate-*r* N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

          \[\leadsto \frac{a + -1 \cdot \frac{\color{blue}{\left(-100 + 1\right)} \cdot a}{{k}^{2}}}{{k}^{2}} \]

       5. distribute-rgt1-in N/A

          \[\leadsto \frac{a + -1 \cdot \frac{\color{blue}{a + -100 \cdot a}}{{k}^{2}}}{{k}^{2}} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}}{{k}^{2}}} \]

   13. Simplified 65.5%

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

   if -1.52 < m < 1.25

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1.25 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot k}, 99, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.66 \cdot 10^{-195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-82}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.66e-195)
     t_0
     (if (<= m 5e-82) a (if (<= m 1.05) t_0 (* a (* (* k k) 99.0)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.66e-195) {
		tmp = t_0;
	} else if (m <= 5e-82) {
		tmp = a;
	} else if (m <= 1.05) {
		tmp = t_0;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-1.66d-195)) then
        tmp = t_0
    else if (m <= 5d-82) then
        tmp = a
    else if (m <= 1.05d0) then
        tmp = t_0
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.66e-195) {
		tmp = t_0;
	} else if (m <= 5e-82) {
		tmp = a;
	} else if (m <= 1.05) {
		tmp = t_0;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -1.66e-195:
		tmp = t_0
	elif m <= 5e-82:
		tmp = a
	elif m <= 1.05:
		tmp = t_0
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.66e-195)
		tmp = t_0;
	elseif (m <= 5e-82)
		tmp = a;
	elseif (m <= 1.05)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -1.66e-195)
		tmp = t_0;
	elseif (m <= 5e-82)
		tmp = a;
	elseif (m <= 1.05)
		tmp = t_0;
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.66e-195], t$95$0, If[LessEqual[m, 5e-82], a, If[LessEqual[m, 1.05], t$95$0, N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.66e-195 or 4.9999999999999998e-82 < m < 1.05000000000000004

   1. Initial program 97.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 56.9

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

   5. Simplified 56.9%

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

   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. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if -1.66e-195 < m < 4.9999999999999998e-82

   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 m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 56.6%

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

   if 1.05000000000000004 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

Alternative 10: 71.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma k (+ k 10.0) 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

   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. *-lowering-*.f64 62.6

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

   8. Simplified 62.6%

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

   if -1.52 < m < 1.1000000000000001

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   if 1.1000000000000001 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.52:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.52)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma k k 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.52) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.52)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.52], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 37.8

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

   5. Simplified 37.8%

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

   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. *-lowering-*.f64 62.6

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

   8. Simplified 62.6%

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

   if -1.52 < m < 1.1000000000000001

   1. Initial program 93.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in k around inf

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

   8. Simplified 89.3%

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

   if 1.1000000000000001 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

Alternative 12: 59.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.8e-168)
   (/ a (* k k))
   (if (<= m 1.05) (/ a (fma k 10.0 1.0)) (* a (* (* k k) 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.8e-168) {
		tmp = a / (k * k);
	} else if (m <= 1.05) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.8e-168)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.05)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.8e-168], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.7999999999999997e-168

   1. Initial program 99.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 53.4

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

   5. Simplified 53.4%

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

   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. *-lowering-*.f64 62.2

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

   8. Simplified 62.2%

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

   if -5.7999999999999997e-168 < m < 1.05000000000000004

   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 89.9

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

   5. Simplified 89.9%

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

   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. accelerator-lowering-fma.f64 64.9

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

   8. Simplified 64.9%

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

   if 1.05000000000000004 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

Alternative 13: 39.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.7:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.7) a (* a (* (* k k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.7) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.7d0) then
        tmp = a
    else
        tmp = a * ((k * k) * 99.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.7) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * 99.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.7:
		tmp = a
	else:
		tmp = a * ((k * k) * 99.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.7)
		tmp = a;
	else
		tmp = Float64(a * Float64(Float64(k * k) * 99.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.7)
		tmp = a;
	else
		tmp = a * ((k * k) * 99.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.7], a, N[(a * N[(N[(k * k), $MachinePrecision] * 99.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.69999999999999996

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 68.8

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

   5. Simplified 68.8%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 27.7%

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

   if 1.69999999999999996 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{0 - \left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)}, a\right) \]

      8. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right), a\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, 0 - \left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right), a\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - \color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}, a\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}, a\right) \]

      13. metadata-eval 29.4

          \[\leadsto \mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, 0 - a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. distribute-rgt-neg-in N/A

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

       13. metadata-eval N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

Alternative 14: 29.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7200:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot -0.0001\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7200.0) a (* a (* (* k k) -0.0001))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 7200.0) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * -0.0001);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7200.0d0) then
        tmp = a
    else
        tmp = a * ((k * k) * (-0.0001d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 7200.0) {
		tmp = a;
	} else {
		tmp = a * ((k * k) * -0.0001);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 7200.0:
		tmp = a
	else:
		tmp = a * ((k * k) * -0.0001)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 7200.0)
		tmp = a;
	else
		tmp = Float64(a * Float64(Float64(k * k) * -0.0001));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 7200.0)
		tmp = a;
	else
		tmp = a * ((k * k) * -0.0001);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 7200.0], a, N[(a * N[(N[(k * k), $MachinePrecision] * -0.0001), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7200

   1. Initial program 95.7%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 68.4

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

   5. Simplified 68.4%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 27.6%

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

   if 7200 < m

   1. Initial program 70.9%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.6

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

   5. Simplified 2.6%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 2.5

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 30.8

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

   10. Simplified 30.8%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{\frac{-1}{10000} \cdot \left(a \cdot {k}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot {k}^{2}\right) \cdot \frac{-1}{10000}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left({k}^{2} \cdot \frac{-1}{10000}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left({k}^{2} \cdot \frac{-1}{10000}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left({k}^{2} \cdot \frac{-1}{10000}\right)} \]

       5. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{-1}{10000}\right) \]

       6. *-lowering-*.f64 27.4

          \[\leadsto a \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot -0.0001\right) \]

   13. Simplified 27.4%

       \[\leadsto \color{blue}{a \cdot \left(\left(k \cdot k\right) \cdot -0.0001\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 24.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{+42}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 4.8e+42) a (* k (* a -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.8e+42) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 4.8d+42) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.8e+42) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 4.8e+42:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 4.8e+42)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 4.8e+42)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 4.8e+42], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.7999999999999997e42

   1. Initial program 93.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 25.9%

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

   if 4.7999999999999997e42 < m

   1. Initial program 72.7%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 19.8

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

   11. Simplified 19.8%

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

Alternative 16: 19.9% 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 88.0%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. /-lowering-/.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. accelerator-lowering-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. +-lowering-+.f64 48.1

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

5. Simplified 48.1%

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

6. Taylor expanded in k around 0

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

8. Simplified 20.2%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))