Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.1%

Time: 11.4s

Alternatives: 14

Speedup: 1.0×

• 22.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.9999999999999999e-154
  2. k = 0.0
  3. m = -2.00000000000000007e154

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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;m \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{a}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= m -2.3e-13)
     (* a (pow k m))
     (if (<= m 5.2e-13)
       (/ (/ a t_0) t_0)
       (* a (/ (pow k m) (* k (/ 1.0 k))))))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (m <= -2.3e-13) {
		tmp = a * pow(k, m);
	} else if (m <= 5.2e-13) {
		tmp = (a / t_0) / t_0;
	} else {
		tmp = a * (pow(k, m) / (k * (1.0 / k)));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (m <= -2.3e-13)
		tmp = Float64(a * (k ^ m));
	elseif (m <= 5.2e-13)
		tmp = Float64(Float64(a / t_0) / t_0);
	else
		tmp = Float64(a * Float64((k ^ m) / Float64(k * Float64(1.0 / k))));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[m, -2.3e-13], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.2e-13], N[(N[(a / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.29999999999999979e-13

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.29999999999999979e-13 < m < 5.2000000000000001e-13

   1. Initial program 89.8%

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

      1. associate-/l* 89.8%

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

      2. remove-double-neg 89.8%

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

      3. distribute-frac-neg2 89.8%

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

      4. distribute-neg-frac2 89.8%

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

      5. remove-double-neg 89.8%

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

      6. sqr-neg 89.8%

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

      7. associate-+l+ 89.8%

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

      8. sqr-neg 89.8%

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

      9. distribute-rgt-out 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in m around 0 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. +-commutative 89.8%

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

      3. +-commutative 89.8%

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

      4. fma-undefine 89.8%

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

      5. add-sqr-sqrt 89.8%

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

      6. times-frac 89.8%

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

      7. fma-undefine 89.8%

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

      8. distribute-lft-in 89.8%

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

      9. associate-+r+ 89.8%

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

      10. +-commutative 89.8%

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

      11. add-sqr-sqrt 89.8%

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

      12. hypot-define 89.8%

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

      13. +-commutative 89.8%

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

      14. fma-define 89.8%

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

      15. fma-undefine 89.8%

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

      16. distribute-lft-in 89.8%

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

      17. associate-+r+ 89.8%

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

      18. +-commutative 89.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

   if 5.2000000000000001e-13 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in k around 0 81.8%

      \[\leadsto a \cdot \frac{{k}^{m}}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in k around inf 81.8%

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

   9. Taylor expanded in k around 0 98.8%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-13}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(1, k\right)}}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.6e-13)
   (* a (pow k m))
   (if (<= m 9.5e-12)
     (/ (/ a (hypot 1.0 k)) (hypot 1.0 k))
     (* a (/ (pow k m) (* k (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.6e-13) {
		tmp = a * pow(k, m);
	} else if (m <= 9.5e-12) {
		tmp = (a / hypot(1.0, k)) / hypot(1.0, k);
	} else {
		tmp = a * (pow(k, m) / (k * (1.0 / k)));
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.6e-13) {
		tmp = a * Math.pow(k, m);
	} else if (m <= 9.5e-12) {
		tmp = (a / Math.hypot(1.0, k)) / Math.hypot(1.0, k);
	} else {
		tmp = a * (Math.pow(k, m) / (k * (1.0 / k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.6e-13:
		tmp = a * math.pow(k, m)
	elif m <= 9.5e-12:
		tmp = (a / math.hypot(1.0, k)) / math.hypot(1.0, k)
	else:
		tmp = a * (math.pow(k, m) / (k * (1.0 / k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.6e-13)
		tmp = Float64(a * (k ^ m));
	elseif (m <= 9.5e-12)
		tmp = Float64(Float64(a / hypot(1.0, k)) / hypot(1.0, k));
	else
		tmp = Float64(a * Float64((k ^ m) / Float64(k * Float64(1.0 / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.6e-13)
		tmp = a * (k ^ m);
	elseif (m <= 9.5e-12)
		tmp = (a / hypot(1.0, k)) / hypot(1.0, k);
	else
		tmp = a * ((k ^ m) / (k * (1.0 / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.6e-13], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9.5e-12], N[(N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.6e-13

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.6e-13 < m < 9.4999999999999995e-12

   1. Initial program 89.8%

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

      1. associate-/l* 89.8%

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

      2. remove-double-neg 89.8%

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

      3. distribute-frac-neg2 89.8%

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

      4. distribute-neg-frac2 89.8%

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

      5. remove-double-neg 89.8%

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

      6. sqr-neg 89.8%

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

      7. associate-+l+ 89.8%

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

      8. sqr-neg 89.8%

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

      9. distribute-rgt-out 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in m around 0 89.8%

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

   6. Taylor expanded in k around inf 88.5%

      \[\leadsto \frac{a}{1 + k \cdot \color{blue}{k}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 88.5%

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

      2. add-sqr-sqrt 88.5%

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

      3. times-frac 88.5%

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

      4. hypot-1-def 88.5%

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

      5. hypot-1-def 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

   10. Applied egg-rr 98.6%

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

   if 9.4999999999999995e-12 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in k around 0 81.8%

      \[\leadsto a \cdot \frac{{k}^{m}}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in k around inf 81.8%

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

   9. Taylor expanded in k around 0 98.8%

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

Alternative 3: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2e-13)
   (* a (pow k m))
   (if (<= m 5.8e-13)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (* a (/ (pow k m) (* k (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-13) {
		tmp = a * pow(k, m);
	} else if (m <= 5.8e-13) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (pow(k, m) / (k * (1.0 / k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-2d-13)) then
        tmp = a * (k ** m)
    else if (m <= 5.8d-13) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * ((k ** m) / (k * (1.0d0 / k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-13) {
		tmp = a * Math.pow(k, m);
	} else if (m <= 5.8e-13) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (Math.pow(k, m) / (k * (1.0 / k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2e-13:
		tmp = a * math.pow(k, m)
	elif m <= 5.8e-13:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (math.pow(k, m) / (k * (1.0 / k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2e-13)
		tmp = Float64(a * (k ^ m));
	elseif (m <= 5.8e-13)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64((k ^ m) / Float64(k * Float64(1.0 / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2e-13)
		tmp = a * (k ^ m);
	elseif (m <= 5.8e-13)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * ((k ^ m) / (k * (1.0 / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2e-13], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.8e-13], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.0000000000000001e-13

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.0000000000000001e-13 < m < 5.7999999999999995e-13

   1. Initial program 89.8%

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

      1. associate-/l* 89.8%

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

      2. remove-double-neg 89.8%

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

      3. distribute-frac-neg2 89.8%

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

      4. distribute-neg-frac2 89.8%

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

      5. remove-double-neg 89.8%

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

      6. sqr-neg 89.8%

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

      7. associate-+l+ 89.8%

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

      8. sqr-neg 89.8%

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

      9. distribute-rgt-out 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in m around 0 89.8%

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

   if 5.7999999999999995e-13 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in k around 0 81.8%

      \[\leadsto a \cdot \frac{{k}^{m}}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in k around inf 81.8%

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

   9. Taylor expanded in k around 0 98.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-13}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \frac{1}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 3.4:\\ \;\;\;\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 3.4) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 3.4) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 3.4d0) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 3.4) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 3.4:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 3.4)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 3.4)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 3.4], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.39999999999999991

   1. Initial program 94.8%

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

   if 3.39999999999999991 < m

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. remove-double-neg 81.1%

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

      3. distribute-frac-neg2 81.1%

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

      4. distribute-neg-frac2 81.1%

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

      5. remove-double-neg 81.1%

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

      6. sqr-neg 81.1%

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

      7. associate-+l+ 81.1%

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

      8. sqr-neg 81.1%

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

      9. distribute-rgt-out 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.4:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.8:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.8) (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0))))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.8) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 4.8d0) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.8) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 4.8:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 4.8)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 4.8)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 4.8], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.79999999999999982

   1. Initial program 94.8%

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

      1. associate-/l* 94.7%

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

      2. remove-double-neg 94.7%

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

      3. distribute-frac-neg2 94.7%

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

      4. distribute-neg-frac2 94.7%

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

      5. remove-double-neg 94.7%

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

      6. sqr-neg 94.7%

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

      7. associate-+l+ 94.7%

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

      8. sqr-neg 94.7%

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

      9. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   if 4.79999999999999982 < m

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. remove-double-neg 81.1%

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

      3. distribute-frac-neg2 81.1%

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

      4. distribute-neg-frac2 81.1%

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

      5. remove-double-neg 81.1%

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

      6. sqr-neg 81.1%

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

      7. associate-+l+ 81.1%

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

      8. sqr-neg 81.1%

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

      9. distribute-rgt-out 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.8:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-15} \lor \neg \left(m \leq 5.5 \cdot 10^{-23}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.35e-15) (not (<= m 5.5e-23)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.35e-15) || !(m <= 5.5e-23)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-1.35d-15)) .or. (.not. (m <= 5.5d-23))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.35e-15) || !(m <= 5.5e-23)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.35e-15) or not (m <= 5.5e-23):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.35e-15) || !(m <= 5.5e-23))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -1.35e-15) || ~((m <= 5.5e-23)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.35e-15], N[Not[LessEqual[m, 5.5e-23]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.35000000000000005e-15 or 5.5000000000000001e-23 < m

   1. Initial program 91.5%

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

      1. associate-/l* 91.5%

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

      2. remove-double-neg 91.5%

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

      3. distribute-frac-neg2 91.5%

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

      4. distribute-neg-frac2 91.5%

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

      5. remove-double-neg 91.5%

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

      6. sqr-neg 91.5%

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

      7. associate-+l+ 91.5%

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

      8. sqr-neg 91.5%

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

      9. distribute-rgt-out 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in k around 0 99.4%

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

   if -1.35000000000000005e-15 < m < 5.5000000000000001e-23

   1. Initial program 89.6%

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

      1. associate-/l* 89.5%

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

      2. remove-double-neg 89.5%

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

      3. distribute-frac-neg2 89.5%

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

      4. distribute-neg-frac2 89.5%

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

      5. remove-double-neg 89.5%

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

      6. sqr-neg 89.5%

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

      7. associate-+l+ 89.5%

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

      8. sqr-neg 89.5%

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

      9. distribute-rgt-out 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in m around 0 89.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-15} \lor \neg \left(m \leq 5.5 \cdot 10^{-23}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.95)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (+ a (* a (* k (- (* k 99.0) 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.95d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.95:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.95)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.95)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.95], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999996

   1. Initial program 94.8%

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

      1. associate-/l* 94.7%

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

      2. remove-double-neg 94.7%

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

      3. distribute-frac-neg2 94.7%

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

      4. distribute-neg-frac2 94.7%

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

      5. remove-double-neg 94.7%

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

      6. sqr-neg 94.7%

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

      7. associate-+l+ 94.7%

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

      8. sqr-neg 94.7%

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

      9. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in m around 0 63.3%

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

   if 1.94999999999999996 < m

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. remove-double-neg 81.1%

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

      3. distribute-frac-neg2 81.1%

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

      4. distribute-neg-frac2 81.1%

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

      5. remove-double-neg 81.1%

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

      6. sqr-neg 81.1%

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

      7. associate-+l+ 81.1%

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

      8. sqr-neg 81.1%

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

      9. distribute-rgt-out 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 28.8%

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

   7. Taylor expanded in a around 0 32.5%

      \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.3) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* k (* 99.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (99.0 * (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 <= 2.3d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (99.0d0 * (a * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (99.0 * (a * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.3:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (99.0 * (a * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.3)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(99.0 * Float64(a * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.3)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (99.0 * (a * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.3], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(99.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2999999999999998

   1. Initial program 94.8%

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

      1. associate-/l* 94.7%

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

      2. remove-double-neg 94.7%

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

      3. distribute-frac-neg2 94.7%

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

      4. distribute-neg-frac2 94.7%

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

      5. remove-double-neg 94.7%

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

      6. sqr-neg 94.7%

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

      7. associate-+l+ 94.7%

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

      8. sqr-neg 94.7%

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

      9. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in m around 0 63.3%

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

   if 2.2999999999999998 < m

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. remove-double-neg 81.1%

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

      3. distribute-frac-neg2 81.1%

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

      4. distribute-neg-frac2 81.1%

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

      5. remove-double-neg 81.1%

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

      6. sqr-neg 81.1%

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

      7. associate-+l+ 81.1%

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

      8. sqr-neg 81.1%

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

      9. distribute-rgt-out 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 28.8%

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

   7. Taylor expanded in a around 0 28.8%

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

   8. Taylor expanded in k around inf 28.8%

      \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.2) (/ a (+ 1.0 (* k k))) (+ a (* k (* 99.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (k * (99.0 * (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 <= 2.2d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + (k * (99.0d0 * (a * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (k * (99.0 * (a * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.2:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (k * (99.0 * (a * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(k * Float64(99.0 * Float64(a * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (k * (99.0 * (a * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.2], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(99.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 94.8%

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

      1. associate-/l* 94.7%

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

      2. remove-double-neg 94.7%

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

      3. distribute-frac-neg2 94.7%

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

      4. distribute-neg-frac2 94.7%

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

      5. remove-double-neg 94.7%

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

      6. sqr-neg 94.7%

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

      7. associate-+l+ 94.7%

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

      8. sqr-neg 94.7%

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

      9. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in m around 0 63.3%

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

   6. Taylor expanded in k around inf 62.6%

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

   if 2.2000000000000002 < m

   1. Initial program 81.1%

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

      1. associate-/l* 81.1%

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

      2. remove-double-neg 81.1%

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

      3. distribute-frac-neg2 81.1%

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

      4. distribute-neg-frac2 81.1%

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

      5. remove-double-neg 81.1%

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

      6. sqr-neg 81.1%

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

      7. associate-+l+ 81.1%

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

      8. sqr-neg 81.1%

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

      9. distribute-rgt-out 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 28.8%

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

   7. Taylor expanded in a around 0 28.8%

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

   8. Taylor expanded in k around inf 28.8%

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

Alternative 10: 50.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.4d+18) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4e+18:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.4e+18)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4e+18], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4e18

   1. Initial program 94.3%

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

      1. associate-/l* 94.3%

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

      2. remove-double-neg 94.3%

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

      3. distribute-frac-neg2 94.3%

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

      4. distribute-neg-frac2 94.3%

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

      5. remove-double-neg 94.3%

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

      6. sqr-neg 94.3%

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

      7. associate-+l+ 94.3%

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

      8. sqr-neg 94.3%

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

      9. distribute-rgt-out 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in m around 0 62.3%

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

   6. Taylor expanded in k around inf 61.6%

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

   if 1.4e18 < m

   1. Initial program 81.7%

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

      1. associate-/l* 81.7%

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

      2. remove-double-neg 81.7%

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

      3. distribute-frac-neg2 81.7%

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

      4. distribute-neg-frac2 81.7%

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

      5. remove-double-neg 81.7%

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

      6. sqr-neg 81.7%

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

      7. associate-+l+ 81.7%

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

      8. sqr-neg 81.7%

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

      9. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 8.5%

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

   7. Taylor expanded in k around inf 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*r* 23.3%

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

      3. *-commutative 23.3%

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

   9. Simplified 23.3%

      \[\leadsto \color{blue}{\left(k \cdot -10\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

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

Alternative 11: 34.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.4d+18) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4e+18:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.4e+18)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4e+18], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4e18

   1. Initial program 94.3%

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

      1. associate-/l* 94.3%

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

      2. remove-double-neg 94.3%

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

      3. distribute-frac-neg2 94.3%

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

      4. distribute-neg-frac2 94.3%

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

      5. remove-double-neg 94.3%

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

      6. sqr-neg 94.3%

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

      7. associate-+l+ 94.3%

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

      8. sqr-neg 94.3%

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

      9. distribute-rgt-out 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in m around 0 62.3%

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

   6. Taylor expanded in k around 0 43.9%

      \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
   7. Step-by-step derivation

      1. *-commutative 82.5%

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

   8. Simplified 43.9%

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

   if 1.4e18 < m

   1. Initial program 81.7%

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

      1. associate-/l* 81.7%

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

      2. remove-double-neg 81.7%

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

      3. distribute-frac-neg2 81.7%

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

      4. distribute-neg-frac2 81.7%

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

      5. remove-double-neg 81.7%

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

      6. sqr-neg 81.7%

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

      7. associate-+l+ 81.7%

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

      8. sqr-neg 81.7%

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

      9. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 8.5%

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

   7. Taylor expanded in k around inf 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*r* 23.3%

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

      3. *-commutative 23.3%

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

   9. Simplified 23.3%

      \[\leadsto \color{blue}{\left(k \cdot -10\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

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

Alternative 12: 25.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.4e+18) a (* a (* k -10.0))))

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.4d+18) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.4e+18) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4e+18:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.4e+18)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.4e+18)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4e+18], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4e18

   1. Initial program 94.3%

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

      1. associate-/l* 94.3%

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

      2. remove-double-neg 94.3%

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

      3. distribute-frac-neg2 94.3%

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

      4. distribute-neg-frac2 94.3%

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

      5. remove-double-neg 94.3%

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

      6. sqr-neg 94.3%

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

      7. associate-+l+ 94.3%

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

      8. sqr-neg 94.3%

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

      9. distribute-rgt-out 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in m around 0 62.3%

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

   6. Taylor expanded in k around 0 30.9%

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

   if 1.4e18 < m

   1. Initial program 81.7%

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

      1. associate-/l* 81.7%

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

      2. remove-double-neg 81.7%

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

      3. distribute-frac-neg2 81.7%

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

      4. distribute-neg-frac2 81.7%

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

      5. remove-double-neg 81.7%

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

      6. sqr-neg 81.7%

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

      7. associate-+l+ 81.7%

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

      8. sqr-neg 81.7%

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

      9. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 8.5%

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

   7. Taylor expanded in k around inf 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*r* 23.3%

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

      3. *-commutative 23.3%

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

   9. Simplified 23.3%

      \[\leadsto \color{blue}{\left(k \cdot -10\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

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

Alternative 13: 25.4% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.4e+18) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.4e+18) {
		tmp = 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 <= 1.4d+18) then
        tmp = 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 <= 1.4e+18) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4e+18:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.4e+18)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.4e+18)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4e+18], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4e18

   1. Initial program 94.3%

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

      1. associate-/l* 94.3%

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

      2. remove-double-neg 94.3%

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

      3. distribute-frac-neg2 94.3%

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

      4. distribute-neg-frac2 94.3%

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

      5. remove-double-neg 94.3%

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

      6. sqr-neg 94.3%

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

      7. associate-+l+ 94.3%

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

      8. sqr-neg 94.3%

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

      9. distribute-rgt-out 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in m around 0 62.3%

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

   6. Taylor expanded in k around 0 30.9%

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

   if 1.4e18 < m

   1. Initial program 81.7%

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

      1. associate-/l* 81.7%

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

      2. remove-double-neg 81.7%

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

      3. distribute-frac-neg2 81.7%

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

      4. distribute-neg-frac2 81.7%

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

      5. remove-double-neg 81.7%

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

      6. sqr-neg 81.7%

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

      7. associate-+l+ 81.7%

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

      8. sqr-neg 81.7%

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

      9. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around 0 8.5%

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

   7. Taylor expanded in k around inf 23.3%

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

      1. *-commutative 23.3%

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

   9. Simplified 23.3%

      \[\leadsto \color{blue}{-10 \cdot \left(k \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

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

Alternative 14: 19.8% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, k, m)
	return a
end

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.8%

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

   1. associate-/l* 90.8%

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

   2. remove-double-neg 90.8%

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

   3. distribute-frac-neg2 90.8%

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

   4. distribute-neg-frac2 90.8%

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

   5. remove-double-neg 90.8%

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

   6. sqr-neg 90.8%

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

   7. associate-+l+ 90.8%

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

   8. sqr-neg 90.8%

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

   9. distribute-rgt-out 90.8%

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

3. Simplified 90.8%

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

5. Taylor expanded in m around 0 45.9%

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

6. Taylor expanded in k around 0 23.4%

   \[\leadsto \color{blue}{a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))