Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.6%

Time: 10.3s

Alternatives: 13

Speedup: 1.1×

• 21.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.06:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.06)
   (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
   (/ a (pow k (- m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.06) {
		tmp = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = a / pow(k, -m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.06d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    else
        tmp = a / (k ** -m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.06) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = a / Math.pow(k, -m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.06:
		tmp = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	else:
		tmp = a / math.pow(k, -m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.06)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)));
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.06)
		tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	else
		tmp = a / (k ^ -m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.06], 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], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.059999999999999998

   1. Initial program 98.3%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. rem-exp-log N/A

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

      14. diff-log N/A

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

      15. lift-log1p.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right)} - \log \left({k}^{m} \cdot a\right)}} \]

      16. lift-log.f64 N/A

          \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \color{blue}{\log \left({k}^{m} \cdot a\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

      18. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

   6. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-flip N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-neg.f64 68.7

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. exp-to-pow N/A

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

       7. lower-pow.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 2: 43.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(99, k, -10\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e-312)
   (/ a (* k k))
   (fma (* a (fma 99.0 k -10.0)) k a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e-312) {
		tmp = a / (k * k);
	} else {
		tmp = fma((a * fma(99.0, k, -10.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e-312)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(a * fma(99.0, k, -10.0)), k, a);
	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[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-312], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] * k + a), $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))) < 2.0000000000019e-312

   1. Initial program 98.3%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 36.5%

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

   if 2.0000000000019e-312 < (/.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 77.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.4%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 53.7%

       \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{\mathsf{fma}\left({k}^{-1}, \frac{99}{k} - 10, 1\right)}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.43)
   (* (/ (fma (pow k -1.0) (- (/ 99.0 k) 10.0) 1.0) (* k k)) a)
   (if (<= m 0.06)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.43) {
		tmp = (fma(pow(k, -1.0), ((99.0 / k) - 10.0), 1.0) / (k * k)) * a;
	} else if (m <= 0.06) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.43)
		tmp = Float64(Float64(fma((k ^ -1.0), Float64(Float64(99.0 / k) - 10.0), 1.0) / Float64(k * k)) * a);
	elseif (m <= 0.06)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.43], N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] * N[(N[(99.0 / k), $MachinePrecision] - 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.06], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.429999999999999993

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 34.3

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

   7. Applied rewrites 34.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 63.8%

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

   if -0.429999999999999993 < m < 0.059999999999999998

   1. Initial program 97.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 34.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{\mathsf{fma}\left({k}^{-1}, \frac{99}{k} - 10, 1\right)}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.06:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.06)
   (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a)
   (/ a (pow k (- m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.06) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = a / pow(k, -m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.06)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.06], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.059999999999999998

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.2

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. rem-exp-log N/A

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

      14. diff-log N/A

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

      15. lift-log1p.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right)} - \log \left({k}^{m} \cdot a\right)}} \]

      16. lift-log.f64 N/A

          \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \color{blue}{\log \left({k}^{m} \cdot a\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

      18. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

   6. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-flip N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-neg.f64 68.7

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. exp-to-pow N/A

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

       7. lower-pow.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.019:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{elif}\;m \leq 10^{-6}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.019)
   (* (pow k m) a)
   (if (<= m 1e-6) (/ a (fma (+ 10.0 k) k 1.0)) (/ a (pow k (- m))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.019) {
		tmp = pow(k, m) * a;
	} else if (m <= 1e-6) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = a / pow(k, -m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.019)
		tmp = Float64((k ^ m) * a);
	elseif (m <= 1e-6)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.019], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 1e-6], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0189999999999999995

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.0189999999999999995 < m < 9.99999999999999955e-7

   1. Initial program 97.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

   if 9.99999999999999955e-7 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. rem-exp-log N/A

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

      14. diff-log N/A

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

      15. lift-log1p.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right)} - \log \left({k}^{m} \cdot a\right)}} \]

      16. lift-log.f64 N/A

          \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \color{blue}{\log \left({k}^{m} \cdot a\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

      18. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(\left(k + 10\right) \cdot k\right) - \log \left({k}^{m} \cdot a\right)}}} \]

   6. Applied rewrites 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-flip N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-neg.f64 68.7

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. exp-to-pow N/A

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

       7. lower-pow.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 6: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.019 \lor \neg \left(m \leq 10^{-6}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.019) (not (<= m 1e-6)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.019) || !(m <= 1e-6)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.019) || !(m <= 1e-6))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.019], N[Not[LessEqual[m, 1e-6]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0189999999999999995 or 9.99999999999999955e-7 < m

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.5

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.0189999999999999995 < m < 9.99999999999999955e-7

   1. Initial program 97.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.019 \lor \neg \left(m \leq 10^{-6}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, \frac{99}{k} - 10, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.43)
   (/ (fma (/ a k) (- (/ 99.0 k) 10.0) a) (* k k))
   (if (<= m 0.06)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.43) {
		tmp = fma((a / k), ((99.0 / k) - 10.0), a) / (k * k);
	} else if (m <= 0.06) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.43)
		tmp = Float64(fma(Float64(a / k), Float64(Float64(99.0 / k) - 10.0), a) / Float64(k * k));
	elseif (m <= 0.06)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.43], N[(N[(N[(a / k), $MachinePrecision] * N[(N[(99.0 / k), $MachinePrecision] - 10.0), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.06], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 58.3%

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

   if -0.429999999999999993 < m < 0.059999999999999998

   1. Initial program 97.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 34.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.43)
   (/ a (* k k))
   (if (<= m 0.06)
     (/ a (fma (+ 10.0 k) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.43) {
		tmp = a / (k * k);
	} else if (m <= 0.06) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.43)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.06)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.43], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.06], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 53.9%

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

   if -0.429999999999999993 < m < 0.059999999999999998

   1. Initial program 97.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.1%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 34.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.1e-16)
   (/ a (* k k))
   (if (<= m 0.06)
     (/ a (fma 10.0 k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.1e-16) {
		tmp = a / (k * k);
	} else if (m <= 0.06) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.1e-16)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.06)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.1e-16], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.06], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.1e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 54.5%

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

   if -1.1e-16 < m < 0.059999999999999998

   1. Initial program 97.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 71.6%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 34.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.06:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(99, k, -10\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.1e-16)
   (/ a (* k k))
   (if (<= m 0.06) (/ a (fma 10.0 k 1.0)) (fma (* a (fma 99.0 k -10.0)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.1e-16) {
		tmp = a / (k * k);
	} else if (m <= 0.06) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = fma((a * fma(99.0, k, -10.0)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.1e-16)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.06)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = fma(Float64(a * fma(99.0, k, -10.0)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.1e-16], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.06], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.1e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 54.5%

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

   if -1.1e-16 < m < 0.059999999999999998

   1. Initial program 97.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 71.6%

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

   if 0.059999999999999998 < m

   1. Initial program 77.1%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 29.8%

       \[\leadsto \mathsf{fma}\left(a \cdot \mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{-296} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -9e-296) (not (<= k 0.1))) (/ a (* k k)) (fma (* -10.0 a) k a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9e-296) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = fma((-10.0 * a), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -9e-296) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(-10.0 * a), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -9e-296], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -9.0000000000000003e-296 or 0.10000000000000001 < k

   1. Initial program 85.4%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 47.6%

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

   if -9.0000000000000003e-296 < 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. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.6%

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

   10. Applied rewrites 56.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{-296} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.5) (fma (* -10.0 a) k a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.5) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.5)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.5], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.5

   1. Initial program 98.3%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 34.7%

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

   if 0.5 < m

   1. Initial program 76.8%

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

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. associate-+l+ N/A

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

      14. distribute-lft1-in N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.3%

      \[\leadsto \mathsf{fma}\left(a \cdot k, \color{blue}{-10}, a\right) \]

   9. Taylor expanded in k around inf

      \[\leadsto -10 \cdot \left(a \cdot \color{blue}{k}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 17.8%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]
   12. Step-by-step derivation

   13. Applied rewrites 17.8%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 8.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(-10 \cdot a\right) \cdot k \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* (* -10.0 a) k))

C

double code(double a, double k, double m) {
	return (-10.0 * a) * k;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = ((-10.0d0) * a) * k
end function

Java

public static double code(double a, double k, double m) {
	return (-10.0 * a) * k;
}

Python

def code(a, k, m):
	return (-10.0 * a) * k

Julia

function code(a, k, m)
	return Float64(Float64(-10.0 * a) * k)
end

MATLAB

function tmp = code(a, k, m)
	tmp = (-10.0 * a) * k;
end

Wolfram

code[a_, k_, m_] := N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]

Derivation

1. Initial program 91.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. associate-+r+ N/A

      \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot k\right) + {k}^{2}}} \]

   3. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{{k}^{2} + \left(1 + 10 \cdot k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{{k}^{2} + \color{blue}{\left(10 \cdot k + 1\right)}} \]

   5. associate-+l+ N/A

      \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + 10 \cdot k\right) + 1}} \]

   6. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot k + {k}^{2}\right)} + 1} \]

   7. associate-+l+ N/A

      \[\leadsto \frac{a}{\color{blue}{10 \cdot k + \left({k}^{2} + 1\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot 1\right)} \cdot k + \left({k}^{2} + 1\right)} \]

   9. lft-mult-inverse N/A

      \[\leadsto \frac{a}{\left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)}\right) \cdot k + \left({k}^{2} + 1\right)} \]

   10. associate-*l* N/A

       \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot k\right)} \cdot k + \left({k}^{2} + 1\right)} \]

   11. associate-*r* N/A

       \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot \left(k \cdot k\right)} + \left({k}^{2} + 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{a}{\left(10 \cdot \frac{1}{k}\right) \cdot \color{blue}{{k}^{2}} + \left({k}^{2} + 1\right)} \]

   13. associate-+l+ N/A

       \[\leadsto \frac{a}{\color{blue}{\left(\left(10 \cdot \frac{1}{k}\right) \cdot {k}^{2} + {k}^{2}\right) + 1}} \]

   14. distribute-lft1-in N/A

       \[\leadsto \frac{a}{\color{blue}{\left(10 \cdot \frac{1}{k} + 1\right) \cdot {k}^{2}} + 1} \]

   15. +-commutative N/A

       \[\leadsto \frac{a}{\color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)} \cdot {k}^{2} + 1} \]

   16. unpow2 N/A

       \[\leadsto \frac{a}{\left(1 + 10 \cdot \frac{1}{k}\right) \cdot \color{blue}{\left(k \cdot k\right)} + 1} \]

   17. associate-*r* N/A

       \[\leadsto \frac{a}{\color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right) \cdot k} + 1} \]

   18. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, k, 1\right)}} \]

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
7. Step-by-step derivation

8. Applied rewrites 24.9%

   \[\leadsto \mathsf{fma}\left(a \cdot k, \color{blue}{-10}, a\right) \]

9. Taylor expanded in k around inf

   \[\leadsto -10 \cdot \left(a \cdot \color{blue}{k}\right) \]
10. Step-by-step derivation

11. Applied rewrites 7.2%

    \[\leadsto \left(a \cdot k\right) \cdot -10 \]
12. Step-by-step derivation

13. Applied rewrites 7.2%

    \[\leadsto \left(-10 \cdot a\right) \cdot k \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))