Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.1%

Time: 6.3s

Alternatives: 13

Speedup: 1.1×

• 21.4% 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: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.098:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.098)
   (* (* (pow k m) a) (fma -10.0 k 1.0))
   (* (pow k (+ -1.0 m)) (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.098) {
		tmp = (pow(k, m) * a) * fma(-10.0, k, 1.0);
	} else {
		tmp = pow(k, (-1.0 + m)) * (a / k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.098)
		tmp = Float64(Float64((k ^ m) * a) * fma(-10.0, k, 1.0));
	else
		tmp = Float64((k ^ Float64(-1.0 + m)) * Float64(a / k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.098], N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] * N[(-10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, N[(-1.0 + m), $MachinePrecision]], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.098000000000000004

   1. Initial program 95.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

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

      8. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. exp-prod N/A

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

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

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 0.098000000000000004 < k

   1. Initial program 82.0%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. exp-prod N/A

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

      7. neg-mul-1 N/A

         \[\leadsto \frac{a}{{k}^{2}} \cdot {\left(e^{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}}\right)}^{m} \]

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

      10. rem-exp-log N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 96.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.098:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(-1 + m\right)} \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 41.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, 10, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k)))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 2e+306)
       (* (fma -10.0 k 1.0) a)
       (if (<= t_1 INFINITY) t_0 (* (fma k 10.0 1.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (pow(k, m) * a) / (((10.0 * k) + 1.0) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+306) {
		tmp = fma(-10.0, k, 1.0) * a;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(k, 10.0, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64((k ^ m) * a) / Float64(Float64(Float64(10.0 * k) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+306)
		tmp = Float64(fma(-10.0, k, 1.0) * a);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(fma(k, 10.0, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+306], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(k * 10.0 + 1.0), $MachinePrecision] * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0 or 2.00000000000000003e306 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 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. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 42.3%

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 42.2%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000003e306

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 98.0%

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

   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 75.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 1.6%

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

   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 24.4%

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

   10. Applied rewrites 30.7%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \mathbf{elif}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, 10, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 16.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(10 \cdot k + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(k, 10, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* (pow k m) a) (+ (+ (* 10.0 k) 1.0) (* k k))) 0.0)
   (* (* -10.0 a) k)
   (* (fma k 10.0 1.0) a)))

C

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(10.0 * k), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision], N[(N[(k * 10.0 + 1.0), $MachinePrecision] * 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))) < 0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 48.7%

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

   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 17.5%

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

   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 8.6%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]

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

   1. Initial program 78.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 43.2%

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

   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 38.6%

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

   10. Applied rewrites 41.5%

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

4. Final simplification 18.9%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-10 \cdot k\right) \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{if}\;m \leq -6.2 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -0.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{k}, 10 - \frac{99}{k}, -1\right)}{k \cdot k} \cdot \left(-a\right)\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (* -10.0 k) (* (pow k m) a))))
   (if (<= m -6.2e+63)
     t_0
     (if (<= m -0.7)
       (* (/ (fma (/ 1.0 k) (- 10.0 (/ 99.0 k)) -1.0) (* k k)) (- a))
       (if (<= m 1.15) (/ a (fma (+ 10.0 k) k 1.0)) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = (-10.0 * k) * (pow(k, m) * a);
	double tmp;
	if (m <= -6.2e+63) {
		tmp = t_0;
	} else if (m <= -0.7) {
		tmp = (fma((1.0 / k), (10.0 - (99.0 / k)), -1.0) / (k * k)) * -a;
	} else if (m <= 1.15) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(-10.0 * k) * Float64((k ^ m) * a))
	tmp = 0.0
	if (m <= -6.2e+63)
		tmp = t_0;
	elseif (m <= -0.7)
		tmp = Float64(Float64(fma(Float64(1.0 / k), Float64(10.0 - Float64(99.0 / k)), -1.0) / Float64(k * k)) * Float64(-a));
	elseif (m <= 1.15)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(-10.0 * k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -6.2e+63], t$95$0, If[LessEqual[m, -0.7], N[(N[(N[(N[(1.0 / k), $MachinePrecision] * N[(10.0 - N[(99.0 / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[m, 1.15], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.2000000000000001e63 or 1.1499999999999999 < m

   1. Initial program 87.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

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

      8. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. exp-prod N/A

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

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

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 75.5%

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

   if -6.2000000000000001e63 < m < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 31.5%

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 3.2%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 77.5%

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

   if -0.69999999999999996 < m < 1.1499999999999999

   1. Initial program 93.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{+63}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot \left({k}^{m} \cdot a\right)\\ \mathbf{elif}\;m \leq -0.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{k}, 10 - \frac{99}{k}, -1\right)}{k \cdot k} \cdot \left(-a\right)\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot \left({k}^{m} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.4e-18)
     (* t_0 (fma -10.0 k 1.0))
     (if (<= m 0.000225) (/ a (fma (+ 10.0 k) k 1.0)) (/ t_0 1.0)))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -1.4e-18) {
		tmp = t_0 * fma(-10.0, k, 1.0);
	} else if (m <= 0.000225) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0 / 1.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.4e-18)
		tmp = Float64(t_0 * fma(-10.0, k, 1.0));
	elseif (m <= 0.000225)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(t_0 / 1.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.4e-18], N[(t$95$0 * N[(-10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.000225], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.40000000000000006e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

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

      8. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. exp-prod N/A

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

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

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -1.40000000000000006e-18 < m < 2.2499999999999999e-4

   1. Initial program 93.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 93.4%

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

   if 2.2499999999999999e-4 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{if}\;m \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (* (pow k m) a) (fma -10.0 k 1.0))))
   (if (<= m -1.4e-18)
     t_0
     (if (<= m 0.000225) (/ a (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (pow(k, m) * a) * fma(-10.0, k, 1.0);
	double tmp;
	if (m <= -1.4e-18) {
		tmp = t_0;
	} else if (m <= 0.000225) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64((k ^ m) * a) * fma(-10.0, k, 1.0))
	tmp = 0.0
	if (m <= -1.4e-18)
		tmp = t_0;
	elseif (m <= 0.000225)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] * N[(-10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.4e-18], t$95$0, If[LessEqual[m, 0.000225], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.40000000000000006e-18 or 2.2499999999999999e-4 < m

   1. Initial program 89.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

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

      8. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. exp-prod N/A

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

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

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 88.9%

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

   if -1.40000000000000006e-18 < m < 2.2499999999999999e-4

   1. Initial program 93.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 93.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \mathsf{fma}\left(-10, k, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{k}, 10 - \frac{99}{k}, -1\right)}{k \cdot k} \cdot \left(-a\right)\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.7)
   (* (/ (fma (/ 1.0 k) (- 10.0 (/ 99.0 k)) -1.0) (* k k)) (- a))
   (if (<= m 1.15) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* -99.0 k) k) (- a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.7) {
		tmp = (fma((1.0 / k), (10.0 - (99.0 / k)), -1.0) / (k * k)) * -a;
	} else if (m <= 1.15) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.7)
		tmp = Float64(Float64(fma(Float64(1.0 / k), Float64(10.0 - Float64(99.0 / k)), -1.0) / Float64(k * k)) * Float64(-a));
	elseif (m <= 1.15)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(-99.0 * k) * k) * Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.7], N[(N[(N[(N[(1.0 / k), $MachinePrecision] * N[(10.0 - N[(99.0 / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[m, 1.15], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 33.7%

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

   7. Applied rewrites 33.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 3.1%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 69.4%

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

   if -0.69999999999999996 < m < 1.1499999999999999

   1. Initial program 93.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.8%

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

   if 1.1499999999999999 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 64.1%

       \[\leadsto \left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{k}, 10 - \frac{99}{k}, -1\right)}{k \cdot k} \cdot \left(-a\right)\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{a - \left(\frac{-99}{k} - -10\right) \cdot \frac{a}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.7)
   (/ (- a (* (- (/ -99.0 k) -10.0) (/ a k))) (* k k))
   (if (<= m 1.15) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* -99.0 k) k) (- a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.7) {
		tmp = (a - (((-99.0 / k) - -10.0) * (a / k))) / (k * k);
	} else if (m <= 1.15) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.7)
		tmp = Float64(Float64(a - Float64(Float64(Float64(-99.0 / k) - -10.0) * Float64(a / k))) / Float64(k * k));
	elseif (m <= 1.15)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(-99.0 * k) * k) * Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.7], N[(N[(a - N[(N[(N[(-99.0 / k), $MachinePrecision] - -10.0), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 33.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.2%

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

   if -0.69999999999999996 < m < 1.1499999999999999

   1. Initial program 93.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.8%

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

   if 1.1499999999999999 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 64.1%

       \[\leadsto \left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{a - \left(\frac{-99}{k} - -10\right) \cdot \frac{a}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.7)
   (/ a (* k k))
   (if (<= m 1.15) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* -99.0 k) k) (- a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.7) {
		tmp = a / (k * k);
	} else if (m <= 1.15) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.7)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.15)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(-99.0 * k) * k) * Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.7], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 33.7%

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

   7. Applied rewrites 33.7%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 63.9%

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

   if -0.69999999999999996 < m < 1.1499999999999999

   1. Initial program 93.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.8%

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

   if 1.1499999999999999 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 64.1%

       \[\leadsto \left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.7:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8e-5)
   (/ a (* k k))
   (if (<= m 1.15) (/ a (fma 10.0 k 1.0)) (* (* (* -99.0 k) k) (- a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8e-5) {
		tmp = a / (k * k);
	} else if (m <= 1.15) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8e-5)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.15)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(-99.0 * k) * k) * Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8e-5], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.00000000000000065e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 33.7%

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

   7. Applied rewrites 33.7%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 63.9%

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

   if -8.00000000000000065e-5 < m < 1.1499999999999999

   1. Initial program 93.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.8%

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

   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 70.3%

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

   if 1.1499999999999999 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 64.1%

       \[\leadsto \left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.8e-252)
   (/ a (* k k))
   (if (<= m 0.38) (* 1.0 a) (* (* (* -99.0 k) k) (- a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.8e-252) {
		tmp = a / (k * k);
	} else if (m <= 0.38) {
		tmp = 1.0 * a;
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.8d-252)) then
        tmp = a / (k * k)
    else if (m <= 0.38d0) then
        tmp = 1.0d0 * a
    else
        tmp = (((-99.0d0) * k) * k) * -a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.8e-252) {
		tmp = a / (k * k);
	} else if (m <= 0.38) {
		tmp = 1.0 * a;
	} else {
		tmp = ((-99.0 * k) * k) * -a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.8e-252:
		tmp = a / (k * k)
	elif m <= 0.38:
		tmp = 1.0 * a
	else:
		tmp = ((-99.0 * k) * k) * -a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.8e-252)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.38)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(-99.0 * k) * k) * Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.8e-252)
		tmp = a / (k * k);
	elseif (m <= 0.38)
		tmp = 1.0 * a;
	else
		tmp = ((-99.0 * k) * k) * -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.8e-252], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.38], N[(1.0 * a), $MachinePrecision], N[(N[(N[(-99.0 * k), $MachinePrecision] * k), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.80000000000000011e-252

   1. Initial program 99.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 50.2%

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

   7. Applied rewrites 50.2%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 61.4%

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

   if -1.80000000000000011e-252 < m < 0.38

   1. Initial program 92.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 92.0%

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

   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 60.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 61.3%

       \[\leadsto 1 \cdot a \]

   if 0.38 < m

   1. Initial program 77.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in k around inf

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

   13. Applied rewrites 64.1%

       \[\leadsto \left(-a\right) \cdot \left(\left(-99 \cdot k\right) \cdot k\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-99 \cdot k\right) \cdot k\right) \cdot \left(-a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65 \cdot 10^{+30}:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.65e+30) (* 1.0 a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.65e+30) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.65d+30) then
        tmp = 1.0d0 * a
    else
        tmp = ((-10.0d0) * a) * k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.65e+30) {
		tmp = 1.0 * a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.65e+30:
		tmp = 1.0 * a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.65e+30)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.65e+30)
		tmp = 1.0 * a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.65e+30], N[(1.0 * a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.65000000000000013e30

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 62.3%

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

   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 29.3%

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

   9. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   10. Step-by-step derivation

   11. Applied rewrites 29.9%

       \[\leadsto 1 \cdot a \]

   if 1.65000000000000013e30 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. 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 2.9%

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

   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 9.2%

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

   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 24.5%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 20.2% accurate, 22.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. lft-mult-inverse N/A

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

   7. associate-*l* N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. 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 47.0%

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

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 24.1%

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

9. Taylor expanded in k around 0

   \[\leadsto 1 \cdot a \]
10. Step-by-step derivation

11. Applied rewrites 23.3%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))