Falkner and Boettcher, Appendix A

Percentage Accurate: 89.6% → 97.5%

Time: 11.3s

Alternatives: 10

Speedup: 1.1×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+226}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 4e+226) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+226) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 4d+226) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+226) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 4e+226:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 4e+226)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 4e+226)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+226], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999985e226

   1. Initial program 97.3%

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

   if 3.99999999999999985e226 < (/.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 57.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, \left(100 - k \cdot k\right) \cdot 0.1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_0 0.0)
     (* a (/ 1.0 (fma k (* (- 100.0 (* k k)) 0.1) 1.0)))
     (if (<= t_0 2e+305) (/ a (fma k (+ k 10.0) 1.0)) (* a (/ 1.0 (* k k)))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a * (1.0 / fma(k, ((100.0 - (k * k)) * 0.1), 1.0));
	} else if (t_0 <= 2e+305) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (1.0 / (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(Float64(100.0 - Float64(k * k)) * 0.1), 1.0)));
	elseif (t_0 <= 2e+305)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a * N[(1.0 / N[(k * N[(N[(100.0 - N[(k * k), $MachinePrecision]), $MachinePrecision] * 0.1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+305], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0

   1. Initial program 96.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.9

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 48.3%

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

      1. flip-+ N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 48.3

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

   9. Applied rewrites 48.3%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(k, \left(100 - k \cdot k\right) \cdot \color{blue}{\frac{1}{10}}, 1\right)} \cdot a \]
   11. Step-by-step derivation

   12. Applied rewrites 46.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(k, \left(100 - k \cdot k\right) \cdot \color{blue}{0.1}, 1\right)} \cdot a \]

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if 1.9999999999999999e305 < (/.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 55.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 55.8

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 22.7

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

   10. Applied rewrites 22.7%

       \[\leadsto \frac{1}{\color{blue}{k \cdot k}} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, \left(100 - k \cdot k\right) \cdot 0.1, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 4e+226)
     (* a (/ (pow k m) (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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 4e+226) {
		tmp = a * (pow(k, m) / 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 (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 4e+226)
		tmp = Float64(a * Float64((k ^ m) / 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[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+226], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999985e226

   1. Initial program 97.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 97.3

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 97.3

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

   4. Applied rewrites 97.3%

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

   if 3.99999999999999985e226 < (/.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 57.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 4 \cdot 10^{+226}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+305)
   (/ a (fma k (+ k 10.0) 1.0))
   (* a (/ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+305) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (1.0 / (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+305)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.9999999999999999e305

   1. Initial program 97.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if 1.9999999999999999e305 < (/.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 55.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 55.8

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-+.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 22.7

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

   10. Applied rewrites 22.7%

       \[\leadsto \frac{1}{\color{blue}{k \cdot k}} \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -5e-17) t_0 (if (<= m 2e-19) (/ 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 <= -5e-17) {
		tmp = t_0;
	} else if (m <= 2e-19) {
		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 <= -5e-17)
		tmp = t_0;
	elseif (m <= 2e-19)
		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, -5e-17], t$95$0, If[LessEqual[m, 2e-19], 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 < -4.9999999999999999e-17 or 2e-19 < m

   1. Initial program 88.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.9999999999999999e-17 < m < 2e-19

   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. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 93.7

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

   5. Applied rewrites 93.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(k, 10 - k, 1\right)\\ \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(t\_0 \cdot 1\right)}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{1 \cdot \left(t\_0 \cdot \frac{-10 - k}{k \cdot \left(k \cdot k\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (fma k (- 10.0 k) 1.0)))
   (if (<= m -36000000000000.0)
     (/ (* a 1.0) (* (* k k) (* t_0 1.0)))
     (if (<= m 1.65)
       (/ a (fma k (+ k 10.0) 1.0))
       (/ (* a 1.0) (* 1.0 (* t_0 (/ (- -10.0 k) (* k (* k k))))))))))

C

double code(double a, double k, double m) {
	double t_0 = fma(k, (10.0 - k), 1.0);
	double tmp;
	if (m <= -36000000000000.0) {
		tmp = (a * 1.0) / ((k * k) * (t_0 * 1.0));
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (a * 1.0) / (1.0 * (t_0 * ((-10.0 - k) / (k * (k * k)))));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = fma(k, Float64(10.0 - k), 1.0)
	tmp = 0.0
	if (m <= -36000000000000.0)
		tmp = Float64(Float64(a * 1.0) / Float64(Float64(k * k) * Float64(t_0 * 1.0)));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(a * 1.0) / Float64(1.0 * Float64(t_0 * Float64(Float64(-10.0 - k) / Float64(k * Float64(k * k))))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(10.0 - k), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[m, -36000000000000.0], N[(N[(a * 1.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 1.0), $MachinePrecision] / N[(1.0 * N[(t$95$0 * N[(N[(-10.0 - k), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e13

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 46.3%

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

   11. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 71.4

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

   13. Applied rewrites 71.4%

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

   if -3.6e13 < m < 1.6499999999999999

   1. Initial program 94.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if 1.6499999999999999 < m

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 3.1%

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

   11. Taylor expanded in k around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. distribute-neg-in N/A

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

       4. metadata-eval N/A

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

       5. unsub-neg N/A

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

       6. metadata-eval N/A

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

       7. *-inverses N/A

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

       8. distribute-frac-neg N/A

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

       9. mul-1-neg N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. div-sub N/A

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

       13. associate-/r* N/A

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

       14. unpow2 N/A

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

       15. cube-mult N/A

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

       16. lower-/.f64 N/A

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

       17. sub-neg N/A

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

       18. metadata-eval N/A

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

       19. +-commutative N/A

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

       20. mul-1-neg N/A

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

       21. unsub-neg N/A

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

       22. lower--.f64 N/A

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

       23. cube-mult N/A

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

       24. unpow2 N/A

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

       25. lower-*.f64 N/A

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

       26. unpow2 N/A

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

       27. lower-*.f64 54.5

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

   13. Applied rewrites 54.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot 1\right)}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{1 \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot \frac{-10 - k}{k \cdot \left(k \cdot k\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(k, 10 - k, 1\right)\\ \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(t\_0 \cdot 1\right)}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{1 \cdot \left(t\_0 \cdot \frac{-1}{k \cdot k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (fma k (- 10.0 k) 1.0)))
   (if (<= m -36000000000000.0)
     (/ (* a 1.0) (* (* k k) (* t_0 1.0)))
     (if (<= m 1.65)
       (/ a (fma k (+ k 10.0) 1.0))
       (/ (* a 1.0) (* 1.0 (* t_0 (/ -1.0 (* k k)))))))))

C

double code(double a, double k, double m) {
	double t_0 = fma(k, (10.0 - k), 1.0);
	double tmp;
	if (m <= -36000000000000.0) {
		tmp = (a * 1.0) / ((k * k) * (t_0 * 1.0));
	} else if (m <= 1.65) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (a * 1.0) / (1.0 * (t_0 * (-1.0 / (k * k))));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = fma(k, Float64(10.0 - k), 1.0)
	tmp = 0.0
	if (m <= -36000000000000.0)
		tmp = Float64(Float64(a * 1.0) / Float64(Float64(k * k) * Float64(t_0 * 1.0)));
	elseif (m <= 1.65)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(a * 1.0) / Float64(1.0 * Float64(t_0 * Float64(-1.0 / Float64(k * k)))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(10.0 - k), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[m, -36000000000000.0], N[(N[(a * 1.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 1.0), $MachinePrecision] / N[(1.0 * N[(t$95$0 * N[(-1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e13

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 46.3%

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

   11. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 71.4

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

   13. Applied rewrites 71.4%

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

   if -3.6e13 < m < 1.6499999999999999

   1. Initial program 94.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if 1.6499999999999999 < m

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 3.1%

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

   11. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 36.1

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

   13. Applied rewrites 36.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot 1\right)}\\ \mathbf{elif}\;m \leq 1.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{1 \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot \frac{-1}{k \cdot k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -36000000000000.0)
   (/ (* a 1.0) (* (* k k) (* (fma k (- 10.0 k) 1.0) 1.0)))
   (/ a (fma k (+ k 10.0) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -36000000000000.0) {
		tmp = (a * 1.0) / ((k * k) * (fma(k, (10.0 - k), 1.0) * 1.0));
	} else {
		tmp = a / fma(k, (k + 10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -36000000000000.0)
		tmp = Float64(Float64(a * 1.0) / Float64(Float64(k * k) * Float64(fma(k, Float64(10.0 - k), 1.0) * 1.0)));
	else
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -36000000000000.0], N[(N[(a * 1.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(k * N[(10.0 - k), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.6e13

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. difference-of-squares N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 46.3%

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

   11. Taylor expanded in k around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 71.4

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

   13. Applied rewrites 71.4%

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

   if -3.6e13 < m

   1. Initial program 86.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 51.2

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

   5. Applied rewrites 51.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -36000000000000:\\ \;\;\;\;\frac{a \cdot 1}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(k, 10 - k, 1\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. lft-mult-inverse N/A

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

   7. associate-*l* N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. distribute-rgt-in N/A

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

   16. associate-*l* N/A

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

   17. lft-mult-inverse N/A

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

   18. metadata-eval N/A

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

   19. *-lft-identity N/A

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

   20. lower-+.f64 45.7

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

5. Applied rewrites 45.7%

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

6. Final simplification 45.7%

   \[\leadsto \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \]
7. Add Preprocessing

Alternative 10: 19.8% accurate, 22.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 90.3

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate-+l+ N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-+.f64 90.3

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in k around 0

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

   1. lower-pow.f64 79.7

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

7. Applied rewrites 79.7%

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

   1. lift-approx N/A

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

   2. lift-*.f64 79.7

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

9. Applied rewrites 18.5%

   \[\leadsto \color{blue}{1 \cdot a} \]

10. Final simplification 18.5%

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

Reproduce

herbie shell --seed 2024212 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))