Falkner and Boettcher, Appendix A

Percentage Accurate: 90.9% → 99.0%

Time: 12.8s

Alternatives: 18

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 3.9609468855270354e+148
  2. k = -1.4239429979188814e-42
  3. m = 6.285891647951671e+176

• 21.0% 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: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.25:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.25) (* a (pow k m)) (/ (* a (pow k (+ m -1.0))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.25) {
		tmp = a * pow(k, m);
	} 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 <= 0.25d0) then
        tmp = a * (k ** m)
    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 <= 0.25) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (a * Math.pow(k, (m + -1.0))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.25:
		tmp = a * math.pow(k, m)
	else:
		tmp = (a * math.pow(k, (m + -1.0))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.25)
		tmp = Float64(a * (k ^ m));
	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 <= 0.25)
		tmp = a * (k ^ m);
	else
		tmp = (a * (k ^ (m + -1.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.25], N[(a * N[Power[k, m], $MachinePrecision]), $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 < 0.25

   1. Initial program 93.8%

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

   3. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 99.0

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

   5. Simplified 99.0%

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

   if 0.25 < k

   1. Initial program 77.6%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.3

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

   5. Simplified 75.3%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      6. /-lowering-/.f64 93.4

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

   7. Applied egg-rr 93.4%

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. div-inv N/A

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

      5. inv-pow N/A

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

      6. pow-prod-up N/A

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

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

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

      8. +-lowering-+.f64 97.5

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

   9. Applied egg-rr 97.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.25:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ (* k 10.0) 1.0)))))
   (if (<= t_1 2e-283)
     t_0
     (if (<= t_1 5e+303)
       (/ a (fma k 10.0 1.0))
       (if (<= t_1 INFINITY) t_0 (fma (* k k) (* a 98.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + ((k * 10.0) + 1.0));
	double tmp;
	if (t_1 <= 2e-283) {
		tmp = t_0;
	} else if (t_1 <= 5e+303) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((k * k), (a * 98.0), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)))
	tmp = 0.0
	if (t_1 <= 2e-283)
		tmp = t_0;
	elseif (t_1 <= 5e+303)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(k * k), Float64(a * 98.0), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-283], t$95$0, If[LessEqual[t$95$1, 5e+303], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(k * k), $MachinePrecision] * N[(a * 98.0), $MachinePrecision] + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.99999999999999989e-283 or 4.9999999999999997e303 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 95.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 40.0

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

   5. Simplified 40.0%

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

   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 38.1

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

   8. Simplified 38.1%

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

   if 1.99999999999999989e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999997e303

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 99.2

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

   5. Simplified 99.2%

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

   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 67.9

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

   8. Simplified 67.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 1.6

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

   5. Simplified 1.6%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 0.0%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + 98 \cdot \left(a \cdot {k}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2} + a} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(98 \cdot a\right)} + a \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left({k}^{2}, 98 \cdot a, a\right)} \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

       8. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ (* k 10.0) 1.0)))))
   (if (<= t_1 2e-283)
     t_0
     (if (<= t_1 5e+303)
       (* a (fma k (fma k 99.0 -10.0) 1.0))
       (if (<= t_1 INFINITY) t_0 (fma (* k k) (* a 98.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + ((k * 10.0) + 1.0));
	double tmp;
	if (t_1 <= 2e-283) {
		tmp = t_0;
	} else if (t_1 <= 5e+303) {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((k * k), (a * 98.0), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)))
	tmp = 0.0
	if (t_1 <= 2e-283)
		tmp = t_0;
	elseif (t_1 <= 5e+303)
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(k * k), Float64(a * 98.0), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-283], t$95$0, If[LessEqual[t$95$1, 5e+303], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(k * k), $MachinePrecision] * N[(a * 98.0), $MachinePrecision] + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.99999999999999989e-283 or 4.9999999999999997e303 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 95.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 40.0

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

   5. Simplified 40.0%

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

   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 38.1

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

   8. Simplified 38.1%

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

   if 1.99999999999999989e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999997e303

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 99.2

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

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 99.1

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 67.6

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

   10. Simplified 67.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 1.6

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

   5. Simplified 1.6%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 0.0%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + 98 \cdot \left(a \cdot {k}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2} + a} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(98 \cdot a\right)} + a \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left({k}^{2}, 98 \cdot a, a\right)} \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

       8. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (* k k) (+ (* k 10.0) 1.0)))))
   (if (<= t_1 2e-283)
     t_0
     (if (<= t_1 5e+303)
       (fma a (* k -10.0) a)
       (if (<= t_1 INFINITY) t_0 (fma (* k k) (* a 98.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((k * k) + ((k * 10.0) + 1.0));
	double tmp;
	if (t_1 <= 2e-283) {
		tmp = t_0;
	} else if (t_1 <= 5e+303) {
		tmp = fma(a, (k * -10.0), a);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((k * k), (a * 98.0), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)))
	tmp = 0.0
	if (t_1 <= 2e-283)
		tmp = t_0;
	elseif (t_1 <= 5e+303)
		tmp = fma(a, Float64(k * -10.0), a);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(k * k), Float64(a * 98.0), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-283], t$95$0, If[LessEqual[t$95$1, 5e+303], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(k * k), $MachinePrecision] * N[(a * 98.0), $MachinePrecision] + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.99999999999999989e-283 or 4.9999999999999997e303 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 95.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 40.0

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

   5. Simplified 40.0%

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

   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 38.1

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

   8. Simplified 38.1%

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

   if 1.99999999999999989e-283 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999997e303

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 99.2

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

   5. Simplified 99.2%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 65.0

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

   8. Simplified 65.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 1.6

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

   5. Simplified 1.6%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 0.0%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + 98 \cdot \left(a \cdot {k}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto a + \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(98 \cdot a\right) \cdot {k}^{2} + a} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(98 \cdot a\right)} + a \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left({k}^{2}, 98 \cdot a, a\right)} \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{k \cdot k}, 98 \cdot a, a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

       8. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(k \cdot k, \color{blue}{a \cdot 98}, a\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, a \cdot 98, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.045:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -1.35e-5)
     t_0
     (if (<= m 0.045) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -1.35e-5) {
		tmp = t_0;
	} else if (m <= 0.045) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -1.35e-5)
		tmp = t_0;
	elseif (m <= 0.045)
		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[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.35e-5], t$95$0, If[LessEqual[m, 0.045], 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 < -1.3499999999999999e-5 or 0.044999999999999998 < m

   1. Initial program 86.6%

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

   3. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.3499999999999999e-5 < m < 0.044999999999999998

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 0.045:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.25:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.25) (* a (pow k m)) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.25) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.25d0) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.25) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.25:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.25)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.25)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.25], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.25

   1. Initial program 93.8%

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

   3. Taylor expanded in k around 0

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

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

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

      2. pow-lowering-pow.f64 99.0

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

   5. Simplified 99.0%

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

   if 0.25 < k

   1. Initial program 77.6%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.3

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

   5. Simplified 75.3%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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 88.3

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

   7. Applied egg-rr 88.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.25:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k \cdot \left(k \cdot k\right)\right)\\ \mathbf{if}\;m \leq -0.23:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{t\_0}, 9603, \mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)\right)}{t\_0}\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right) - \left(k \cdot k\right) \cdot 100}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (* k (* k k)))))
   (if (<= m -0.23)
     (/ (fma (/ a t_0) 9603.0 (fma a (/ 98.0 (* k k)) a)) t_0)
     (if (<= m 1.8)
       (/ a (fma k (+ k 10.0) 1.0))
       (if (<= m 2.4e+87)
         (* a (fma (* k k) (fma k k -1.0) 1.0))
         (/
          a
          (/
           (- (* (fma k k 1.0) (fma k k 1.0)) (* (* k k) 100.0))
           (* k k))))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k * (k * k));
	double tmp;
	if (m <= -0.23) {
		tmp = fma((a / t_0), 9603.0, fma(a, (98.0 / (k * k)), a)) / t_0;
	} else if (m <= 1.8) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else if (m <= 2.4e+87) {
		tmp = a * fma((k * k), fma(k, k, -1.0), 1.0);
	} else {
		tmp = a / (((fma(k, k, 1.0) * fma(k, k, 1.0)) - ((k * k) * 100.0)) / (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k * Float64(k * k)))
	tmp = 0.0
	if (m <= -0.23)
		tmp = Float64(fma(Float64(a / t_0), 9603.0, fma(a, Float64(98.0 / Float64(k * k)), a)) / t_0);
	elseif (m <= 1.8)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	elseif (m <= 2.4e+87)
		tmp = Float64(a * fma(Float64(k * k), fma(k, k, -1.0), 1.0));
	else
		tmp = Float64(a / Float64(Float64(Float64(fma(k, k, 1.0) * fma(k, k, 1.0)) - Float64(Float64(k * k) * 100.0)) / Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.23], N[(N[(N[(a / t$95$0), $MachinePrecision] * 9603.0 + N[(a * N[(98.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[m, 1.8], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4e+87], N[(a * N[(N[(k * k), $MachinePrecision] * N[(k * k + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(N[(N[(k * k + 1.0), $MachinePrecision] * N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(k * k), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.23000000000000001

   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 34.3

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

   5. Simplified 34.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 20.8

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 22.1%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + \left(-1 \cdot \frac{a}{{k}^{4}} + \left(98 \cdot \frac{a}{{k}^{2}} + 9604 \cdot \frac{a}{{k}^{4}}\right)\right)}{{k}^{4}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + \left(-1 \cdot \frac{a}{{k}^{4}} + \left(98 \cdot \frac{a}{{k}^{2}} + 9604 \cdot \frac{a}{{k}^{4}}\right)\right)}{{k}^{4}}} \]

   13. Simplified 85.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, 9603, \mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   if -0.23000000000000001 < m < 1.80000000000000004

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

   if 1.80000000000000004 < m < 2.39999999999999981e87

   1. Initial program 60.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. 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 2.7

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

   7. Applied egg-rr 2.7%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 2.7%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. accelerator-lowering-fma.f64 49.8

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

   13. Simplified 49.8%

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

   if 2.39999999999999981e87 < m

   1. Initial program 81.4%

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

   3. Taylor expanded in 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 3.0

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

   5. Simplified 3.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 2.6

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

   7. Applied egg-rr 2.6%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 40.0

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

   10. Simplified 40.0%

       \[\leadsto \frac{a}{\frac{\left(k \cdot k\right) \cdot 100 - \mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right)}{\color{blue}{k \cdot \left(-k\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.23:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}, 9603, \mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right) - \left(k \cdot k\right) \cdot 100}{k \cdot k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.028:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 9 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right) - \left(k \cdot k\right) \cdot 100}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.028)
   (/ (fma a (/ 98.0 (* k k)) a) (* k (* k (* k k))))
   (if (<= m 2.1)
     (/ a (fma k (+ k 10.0) 1.0))
     (if (<= m 9e+87)
       (* a (fma (* k k) (fma k k -1.0) 1.0))
       (/
        a
        (/ (- (* (fma k k 1.0) (fma k k 1.0)) (* (* k k) 100.0)) (* k k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.028) {
		tmp = fma(a, (98.0 / (k * k)), a) / (k * (k * (k * k)));
	} else if (m <= 2.1) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else if (m <= 9e+87) {
		tmp = a * fma((k * k), fma(k, k, -1.0), 1.0);
	} else {
		tmp = a / (((fma(k, k, 1.0) * fma(k, k, 1.0)) - ((k * k) * 100.0)) / (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.028)
		tmp = Float64(fma(a, Float64(98.0 / Float64(k * k)), a) / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 2.1)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	elseif (m <= 9e+87)
		tmp = Float64(a * fma(Float64(k * k), fma(k, k, -1.0), 1.0));
	else
		tmp = Float64(a / Float64(Float64(Float64(fma(k, k, 1.0) * fma(k, k, 1.0)) - Float64(Float64(k * k) * 100.0)) / Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.028], N[(N[(a * N[(98.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.1], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9e+87], N[(a * N[(N[(k * k), $MachinePrecision] * N[(k * k + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(N[(N[(k * k + 1.0), $MachinePrecision] * N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(k * k), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.0280000000000000006

   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 34.3

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

   5. Simplified 34.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 20.8

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 22.1%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{98 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{4}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{98 \cdot a}{{k}^{2}}} + a}{{k}^{4}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 98}}{{k}^{2}} + a}{{k}^{4}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{98}{{k}^{2}}} + a}{{k}^{4}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{98 \cdot 1}}{{k}^{2}} + a}{{k}^{4}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(98 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{4}} \]

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

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, 98 \cdot \frac{1}{{k}^{2}}, a\right)}}{{k}^{4}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98 \cdot 1}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{98}}{{k}^{2}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       12. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

       15. pow-sqr N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       16. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       17. associate-*l* N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot \left(k \cdot {k}^{2}\right)}} \]

       18. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       19. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{{k}^{3}}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot {k}^{3}}} \]

       21. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

       22. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}} \]

       24. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       25. *-lowering-*.f64 82.5

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. Simplified 82.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   if -0.0280000000000000006 < m < 2.10000000000000009

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

   if 2.10000000000000009 < m < 9.0000000000000005e87

   1. Initial program 60.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   6. 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 2.7

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

   7. Applied egg-rr 2.7%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 2.7%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. accelerator-lowering-fma.f64 49.8

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

   13. Simplified 49.8%

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

   if 9.0000000000000005e87 < m

   1. Initial program 81.4%

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

   3. Taylor expanded in 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 3.0

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

   5. Simplified 3.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 2.6

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

   7. Applied egg-rr 2.6%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 40.0

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

   10. Simplified 40.0%

       \[\leadsto \frac{a}{\frac{\left(k \cdot k\right) \cdot 100 - \mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right)}{\color{blue}{k \cdot \left(-k\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.028:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 9 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k, 1\right) \cdot \mathsf{fma}\left(k, k, 1\right) - \left(k \cdot k\right) \cdot 100}{k \cdot k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.22:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, a \cdot 940996, a \cdot 9603\right), a \cdot 98\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.22)
   (/ (fma a (/ 98.0 (* k k)) a) (* k (* k (* k k))))
   (if (<= m 2.3)
     (/ a (fma k (+ k 10.0) 1.0))
     (fma
      (* k k)
      (fma (* k k) (fma (* k k) (* a 940996.0) (* a 9603.0)) (* a 98.0))
      a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.22) {
		tmp = fma(a, (98.0 / (k * k)), a) / (k * (k * (k * k)));
	} else if (m <= 2.3) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = fma((k * k), fma((k * k), fma((k * k), (a * 940996.0), (a * 9603.0)), (a * 98.0)), a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.22)
		tmp = Float64(fma(a, Float64(98.0 / Float64(k * k)), a) / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 2.3)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = fma(Float64(k * k), fma(Float64(k * k), fma(Float64(k * k), Float64(a * 940996.0), Float64(a * 9603.0)), Float64(a * 98.0)), a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.22], N[(N[(a * N[(98.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(a * 940996.0), $MachinePrecision] + N[(a * 9603.0), $MachinePrecision]), $MachinePrecision] + N[(a * 98.0), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.220000000000000001

   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 34.3

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

   5. Simplified 34.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 20.8

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 22.1%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{98 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{4}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{98 \cdot a}{{k}^{2}}} + a}{{k}^{4}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 98}}{{k}^{2}} + a}{{k}^{4}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{98}{{k}^{2}}} + a}{{k}^{4}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{98 \cdot 1}}{{k}^{2}} + a}{{k}^{4}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(98 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{4}} \]

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

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, 98 \cdot \frac{1}{{k}^{2}}, a\right)}}{{k}^{4}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98 \cdot 1}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{98}}{{k}^{2}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       12. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

       15. pow-sqr N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       16. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       17. associate-*l* N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot \left(k \cdot {k}^{2}\right)}} \]

       18. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       19. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{{k}^{3}}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot {k}^{3}}} \]

       21. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

       22. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}} \]

       24. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       25. *-lowering-*.f64 82.5

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. Simplified 82.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   if -0.220000000000000001 < m < 2.2999999999999998

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

   if 2.2999999999999998 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 2.5

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 2.4%

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

   11. Taylor expanded in k around 0

       \[\leadsto \color{blue}{a + {k}^{2} \cdot \left(98 \cdot a + {k}^{2} \cdot \left(-1 \cdot a + \left(9604 \cdot a + {k}^{2} \cdot \left(-98 \cdot a + 98 \cdot \left(-1 \cdot a + 9604 \cdot a\right)\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{k}^{2} \cdot \left(98 \cdot a + {k}^{2} \cdot \left(-1 \cdot a + \left(9604 \cdot a + {k}^{2} \cdot \left(-98 \cdot a + 98 \cdot \left(-1 \cdot a + 9604 \cdot a\right)\right)\right)\right)\right) + a} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left({k}^{2}, 98 \cdot a + {k}^{2} \cdot \left(-1 \cdot a + \left(9604 \cdot a + {k}^{2} \cdot \left(-98 \cdot a + 98 \cdot \left(-1 \cdot a + 9604 \cdot a\right)\right)\right)\right), a\right)} \]

   13. Simplified 37.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, a \cdot 940996, a \cdot 9603\right), a \cdot 98\right), a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.22:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k \cdot k, a \cdot 940996, a \cdot 9603\right), a \cdot 98\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.13:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.13)
   (/ (fma a (/ 98.0 (* k k)) a) (* k (* k (* k k))))
   (if (<= m 2.2)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (fma (* k k) (fma k k -1.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.13) {
		tmp = fma(a, (98.0 / (k * k)), a) / (k * (k * (k * k)));
	} else if (m <= 2.2) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * fma((k * k), fma(k, k, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.13)
		tmp = Float64(fma(a, Float64(98.0 / Float64(k * k)), a) / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 2.2)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * fma(Float64(k * k), fma(k, k, -1.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.13], N[(N[(a * N[(98.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * N[(k * k + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.13

   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 34.3

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

   5. Simplified 34.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 20.8

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 22.1%

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

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{a + 98 \cdot \frac{a}{{k}^{2}}}{{k}^{4}}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{98 \cdot \frac{a}{{k}^{2}} + a}}{{k}^{4}} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{98 \cdot a}{{k}^{2}}} + a}{{k}^{4}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 98}}{{k}^{2}} + a}{{k}^{4}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{98}{{k}^{2}}} + a}{{k}^{4}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{98 \cdot 1}}{{k}^{2}} + a}{{k}^{4}} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(98 \cdot \frac{1}{{k}^{2}}\right)} + a}{{k}^{4}} \]

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

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, 98 \cdot \frac{1}{{k}^{2}}, a\right)}}{{k}^{4}} \]

       9. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98 \cdot 1}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{\color{blue}{98}}{{k}^{2}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\frac{98}{{k}^{2}}}, a\right)}{{k}^{4}} \]

       12. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{\color{blue}{k \cdot k}}, a\right)}{{k}^{4}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

       15. pow-sqr N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       16. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}} \]

       17. associate-*l* N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot \left(k \cdot {k}^{2}\right)}} \]

       18. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       19. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{{k}^{3}}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{\color{blue}{k \cdot {k}^{3}}} \]

       21. cube-mult N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

       22. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}} \]

       24. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

       25. *-lowering-*.f64 82.5

           \[\leadsto \frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. Simplified 82.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   if -0.13 < m < 2.2000000000000002

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

   if 2.2000000000000002 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 2.9%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. accelerator-lowering-fma.f64 35.8

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

   13. Simplified 35.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.13:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{98}{k \cdot k}, a\right)}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1.25 \cdot 10^{-203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{-306}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a, k \cdot -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -1.25e-203)
     t_0
     (if (<= k 1.15e-306)
       (* a (* k -10.0))
       (if (<= k 0.1) (fma a (* k -10.0) a) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -1.25e-203) {
		tmp = t_0;
	} else if (k <= 1.15e-306) {
		tmp = a * (k * -10.0);
	} else if (k <= 0.1) {
		tmp = fma(a, (k * -10.0), a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -1.25e-203)
		tmp = t_0;
	elseif (k <= 1.15e-306)
		tmp = Float64(a * Float64(k * -10.0));
	elseif (k <= 0.1)
		tmp = fma(a, Float64(k * -10.0), a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.25e-203], t$95$0, If[LessEqual[k, 1.15e-306], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.25e-203 or 0.10000000000000001 < k

   1. Initial program 80.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 45.8

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

   8. Simplified 45.8%

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

   if -1.25e-203 < k < 1.14999999999999995e-306

   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 3.9

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

   5. Simplified 3.9%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 3.9

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

   8. Simplified 3.9%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 46.1

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

   11. Simplified 46.1%

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

   if 1.14999999999999995e-306 < k < 0.10000000000000001

   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 49.5

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

   5. Simplified 49.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 48.9

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

   8. Simplified 48.9%

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

Alternative 12: 71.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.085:\\ \;\;\;\;\frac{a}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.085)
   (/ a (* k (* k (* k k))))
   (if (<= m 2.2)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (fma (* k k) (fma k k -1.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.085) {
		tmp = a / (k * (k * (k * k)));
	} else if (m <= 2.2) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * fma((k * k), fma(k, k, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.085)
		tmp = Float64(a / Float64(k * Float64(k * Float64(k * k))));
	elseif (m <= 2.2)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * fma(Float64(k * k), fma(k, k, -1.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.085], N[(a / N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * N[(k * k + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0850000000000000061

   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 34.3

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

   5. Simplified 34.3%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. flip-+ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. accelerator-lowering-fma.f64 20.8

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 22.1%

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

   11. Taylor expanded in k around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. cube-mult N/A

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

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

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

       8. cube-mult N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 81.1

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

   13. Simplified 81.1%

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

   if -0.0850000000000000061 < m < 2.2000000000000002

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. /-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 88.3

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

   5. Simplified 88.3%

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

   if 2.2000000000000002 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 2.9%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. accelerator-lowering-fma.f64 35.8

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

   13. Simplified 35.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.085:\\ \;\;\;\;\frac{a}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.8e-6)
   (* a (/ 1.0 (* k k)))
   (if (<= m 1.8)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (fma (* k k) (fma k k -1.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.8e-6) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 1.8) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * fma((k * k), fma(k, k, -1.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.8e-6)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 1.8)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * fma(Float64(k * k), fma(k, k, -1.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.8e-6], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.8], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * k), $MachinePrecision] * N[(k * k + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.8e-6

   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 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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 59.4

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

   10. Simplified 59.4%

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

   if -3.8e-6 < m < 1.80000000000000004

   1. Initial program 89.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.9

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

   5. Simplified 88.9%

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

   if 1.80000000000000004 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around inf

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

   10. Simplified 2.9%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. sub-neg N/A

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

       6. unpow2 N/A

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

       7. metadata-eval N/A

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

       8. accelerator-lowering-fma.f64 35.8

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

   13. Simplified 35.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(k, k, -1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.8e-6)
   (* a (/ 1.0 (* k k)))
   (if (<= m 2.2)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (fma k (fma k 99.0 -10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.8e-6) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 2.2) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.8e-6)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 2.2)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.8e-6], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.8e-6

   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 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 100.0

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

   5. Simplified 100.0%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in m around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 59.4

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

   10. Simplified 59.4%

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

   if -3.8e-6 < m < 2.2000000000000002

   1. Initial program 89.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.9

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

   5. Simplified 88.9%

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

   if 2.2000000000000002 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 30.6

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

   10. Simplified 30.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.8e-6)
   (/ a (* k k))
   (if (<= m 1.95)
     (/ a (fma k (+ k 10.0) 1.0))
     (* a (fma k (fma k 99.0 -10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.8e-6) {
		tmp = a / (k * k);
	} else if (m <= 1.95) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.8e-6)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.95)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.8e-6], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.95], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.8e-6

   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 34.3

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

   5. Simplified 34.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 59.4

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

   8. Simplified 59.4%

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

   if -3.8e-6 < m < 1.94999999999999996

   1. Initial program 89.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.9

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

   5. Simplified 88.9%

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

   if 1.94999999999999996 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 30.6

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

   10. Simplified 30.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2550:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2550.0)
   (/ a (* k k))
   (if (<= m 2.3) (/ a (fma k k 1.0)) (* a (fma k (fma k 99.0 -10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2550.0) {
		tmp = a / (k * k);
	} else if (m <= 2.3) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = a * fma(k, fma(k, 99.0, -10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2550.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.3)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(a * fma(k, fma(k, 99.0, -10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2550.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k * 99.0 + -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2550

   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 34.7

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

   5. Simplified 34.7%

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

   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 60.6

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

   8. Simplified 60.6%

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

   if -2550 < m < 2.2999999999999998

   1. Initial program 89.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 87.4

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

   5. Simplified 87.4%

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

   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 83.4%

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

   if 2.2999999999999998 < m

   1. Initial program 76.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
   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 2.9

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 30.6

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

   10. Simplified 30.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2550:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(k, 99, -10\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 2.35e+18) a (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.35e+18) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.35d+18) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.35e+18) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.35e+18:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.35e+18)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.35e+18)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.35e+18], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.35e18

   1. Initial program 93.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in k around 0

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

   8. Simplified 25.1%

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

   if 2.35e18 < m

   1. Initial program 77.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Simplified 2.9%

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

   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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 5.0

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

   8. Simplified 5.0%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 18.9

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

   11. Simplified 18.9%

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

Alternative 18: 20.6% 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 87.8%

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

3. Taylor expanded in m around 0

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

   1. /-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 42.3

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

5. Simplified 42.3%

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

6. Taylor expanded in k around 0

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

8. Simplified 17.3%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))