Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 99.9%

Time: 14.1s

Alternatives: 16

Speedup: 1.0×

• 21.9% 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: 90.0% 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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 6e-10)
   (* a (* (fma k -10.0 1.0) (pow k m)))
   (/ (* (/ -1.0 (- (- (/ -1.0 k) k) 10.0)) (* a (pow k m))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 6e-10) {
		tmp = a * (fma(k, -10.0, 1.0) * pow(k, m));
	} else {
		tmp = ((-1.0 / (((-1.0 / k) - k) - 10.0)) * (a * pow(k, m))) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 6e-10)
		tmp = Float64(a * Float64(fma(k, -10.0, 1.0) * (k ^ m)));
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(Float64(-1.0 / k) - k) - 10.0)) * Float64(a * (k ^ m))) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 6e-10], N[(a * N[(N[(k * -10.0 + 1.0), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[(N[(N[(-1.0 / k), $MachinePrecision] - k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] * N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e-10

   1. Initial program 91.1%

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

      1. associate-/l* 91.1%

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

      2. remove-double-neg 91.1%

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

      3. distribute-frac-neg2 91.1%

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

      4. distribute-neg-frac2 91.1%

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

      5. remove-double-neg 91.1%

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

      6. sqr-neg 91.1%

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

      7. associate-+l+ 91.1%

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

      8. sqr-neg 91.1%

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

      9. distribute-rgt-out 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in k around 0 85.9%

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

      1. associate-*r* 85.9%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-define 100.0%

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

   7. Simplified 100.0%

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

   if 6e-10 < k

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in k around inf 81.5%

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

      1. add-sqr-sqrt 54.5%

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

      2. distribute-lft-out 54.4%

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

      3. times-frac 63.8%

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

      4. associate-+l+ 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. associate-*l/ 63.8%

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

      2. div-inv 63.8%

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

      3. associate-*r* 63.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 3 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\left(\frac{-1}{k} - k\right) - 10} \cdot t\_0}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 3e-25) t_0 (/ (* (/ -1.0 (- (- (/ -1.0 k) k) 10.0)) t_0) k))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 3e-25) {
		tmp = t_0;
	} else {
		tmp = ((-1.0 / (((-1.0 / k) - k) - 10.0)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= 3d-25) then
        tmp = t_0
    else
        tmp = (((-1.0d0) / ((((-1.0d0) / k) - k) - 10.0d0)) * t_0) / k
    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 (k <= 3e-25) {
		tmp = t_0;
	} else {
		tmp = ((-1.0 / (((-1.0 / k) - k) - 10.0)) * t_0) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 3e-25:
		tmp = t_0
	else:
		tmp = ((-1.0 / (((-1.0 / k) - k) - 10.0)) * t_0) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 3e-25)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(Float64(-1.0 / k) - k) - 10.0)) * t_0) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 3e-25)
		tmp = t_0;
	else
		tmp = ((-1.0 / (((-1.0 / k) - k) - 10.0)) * t_0) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3e-25], t$95$0, N[(N[(N[(-1.0 / N[(N[(N[(-1.0 / k), $MachinePrecision] - k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9999999999999998e-25

   1. Initial program 90.9%

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

      1. associate-/l* 90.9%

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

      2. remove-double-neg 90.9%

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

      3. distribute-frac-neg2 90.9%

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

      4. distribute-neg-frac2 90.9%

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

      5. remove-double-neg 90.9%

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

      6. sqr-neg 90.9%

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

      7. associate-+l+ 90.9%

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

      8. sqr-neg 90.9%

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

      9. distribute-rgt-out 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 2.9999999999999998e-25 < k

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. add-sqr-sqrt 55.4%

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

      2. distribute-lft-out 55.4%

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

      3. times-frac 64.3%

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

      4. associate-+l+ 64.3%

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

   7. Applied egg-rr 64.3%

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

      1. associate-*l/ 63.9%

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

      2. div-inv 63.9%

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

      3. associate-*r* 64.0%

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

      4. add-sqr-sqrt 99.5%

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

      5. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-25}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\left(\frac{-1}{k} - k\right) - 10} \cdot \left(a \cdot {k}^{m}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-25}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{10 + \left(k + \frac{1}{k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5e-25)
   (* a (pow k m))
   (* (/ (pow k m) k) (/ a (+ 10.0 (+ k (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-25) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) / k) * (a / (10.0 + (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 (k <= 5d-25) then
        tmp = a * (k ** m)
    else
        tmp = ((k ** m) / k) * (a / (10.0d0 + (k + (1.0d0 / k))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-25) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) / k) * (a / (10.0 + (k + (1.0 / k))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 5e-25:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) / k) * (a / (10.0 + (k + (1.0 / k))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5e-25)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) / k) * Float64(a / Float64(10.0 + Float64(k + Float64(1.0 / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 5e-25)
		tmp = a * (k ^ m);
	else
		tmp = ((k ^ m) / k) * (a / (10.0 + (k + (1.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5e-25], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a / N[(10.0 + N[(k + N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999962e-25

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. remove-double-neg 91.0%

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

      3. distribute-frac-neg2 91.0%

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

      4. distribute-neg-frac2 91.0%

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

      5. remove-double-neg 91.0%

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

      6. sqr-neg 91.0%

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

      7. associate-+l+ 91.0%

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

      8. sqr-neg 91.0%

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

      9. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 4.99999999999999962e-25 < k

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in k around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. distribute-lft-out 82.1%

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

      3. times-frac 97.2%

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

      4. associate-+l+ 97.2%

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-25}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{10 + \left(k + \frac{1}{k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-26}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k + \left(10 - \frac{-1}{k}\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4e-26)
   (* a (pow k m))
   (/ (* (pow k m) (/ a (+ k (- 10.0 (/ -1.0 k))))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4e-26) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) * (a / (k + (10.0 - (-1.0 / k))))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 4d-26) then
        tmp = a * (k ** m)
    else
        tmp = ((k ** m) * (a / (k + (10.0d0 - ((-1.0d0) / k))))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4e-26) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) * (a / (k + (10.0 - (-1.0 / k))))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4e-26:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) * (a / (k + (10.0 - (-1.0 / k))))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4e-26)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) * Float64(a / Float64(k + Float64(10.0 - Float64(-1.0 / k))))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4e-26)
		tmp = a * (k ^ m);
	else
		tmp = ((k ^ m) * (a / (k + (10.0 - (-1.0 / k))))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4e-26], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(k + N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.0000000000000002e-26

   1. Initial program 90.9%

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

      1. associate-/l* 90.9%

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

      2. remove-double-neg 90.9%

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

      3. distribute-frac-neg2 90.9%

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

      4. distribute-neg-frac2 90.9%

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

      5. remove-double-neg 90.9%

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

      6. sqr-neg 90.9%

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

      7. associate-+l+ 90.9%

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

      8. sqr-neg 90.9%

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

      9. distribute-rgt-out 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 4.0000000000000002e-26 < k

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in k around inf 82.3%

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

      1. add-sqr-sqrt 55.4%

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

      2. distribute-lft-out 55.4%

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

      3. times-frac 64.3%

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

      4. associate-+l+ 64.3%

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

   7. Applied egg-rr 64.3%

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

   8. Taylor expanded in a around 0 82.3%

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

      1. *-commutative 82.3%

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

      2. times-frac 97.2%

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

      3. associate-*r/ 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-+l+ 97.2%

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

      6. +-commutative 97.2%

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

   10. Simplified 97.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-26}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k + \left(10 - \frac{-1}{k}\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 3.8:\\ \;\;\;\;\frac{t\_0}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= m 3.8) (/ t_0 (+ (* k k) 1.0)) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 3.8) {
		tmp = t_0 / ((k * k) + 1.0);
	} 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 <= 3.8d0) then
        tmp = t_0 / ((k * k) + 1.0d0)
    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 <= 3.8) {
		tmp = t_0 / ((k * k) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 3.8:
		tmp = t_0 / ((k * k) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 3.8)
		tmp = Float64(t_0 / Float64(Float64(k * k) + 1.0));
	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 <= 3.8)
		tmp = t_0 / ((k * k) + 1.0);
	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, 3.8], N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7999999999999998

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in k around inf 95.8%

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

   6. Taylor expanded in k around 0 94.1%

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

   if 3.7999999999999998 < m

   1. Initial program 73.0%

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

      1. associate-/l* 73.0%

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

      2. remove-double-neg 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-neg-frac2 73.0%

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

      5. remove-double-neg 73.0%

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

      6. sqr-neg 73.0%

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

      7. associate-+l+ 73.0%

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

      8. sqr-neg 73.0%

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

      9. distribute-rgt-out 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.8:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k} \cdot t\_0}{k}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 1.0) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / k) * t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= 1.0d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / k) * t_0) / k
    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 (k <= 1.0) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / k) * t_0) / k;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 91.3%

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

      1. associate-/l* 91.3%

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

      2. remove-double-neg 91.3%

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

      3. distribute-frac-neg2 91.3%

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

      4. distribute-neg-frac2 91.3%

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

      5. remove-double-neg 91.3%

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

      6. sqr-neg 91.3%

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

      7. associate-+l+ 91.3%

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

      8. sqr-neg 91.3%

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

      9. distribute-rgt-out 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in k around 0 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   if 1 < k

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in k around inf 80.6%

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

      1. add-sqr-sqrt 54.7%

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

      2. distribute-lft-out 54.7%

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

      3. times-frac 64.5%

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

      4. associate-+l+ 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. associate-*l/ 64.5%

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

      2. div-inv 64.5%

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

      3. associate-*r* 64.6%

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

      4. add-sqr-sqrt 99.9%

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

      5. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in k around inf 98.1%

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

4. Final simplification 98.8%

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

Alternative 7: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{-9} \lor \neg \left(m \leq 0.015\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.55e-9) (not (<= m 0.015)))
   (* a (pow k m))
   (/ a (+ (* k (+ k 10.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.55e-9) || !(m <= 0.015)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	}
	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 <= (-1.55d-9)) .or. (.not. (m <= 0.015d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.55e-9) || !(m <= 0.015)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.55e-9) or not (m <= 0.015):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.55e-9) || !(m <= 0.015))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -1.55e-9) || ~((m <= 0.015)))
		tmp = a * (k ^ m);
	else
		tmp = a / ((k * (k + 10.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.55e-9], N[Not[LessEqual[m, 0.015]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.55000000000000002e-9 or 0.014999999999999999 < m

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. remove-double-neg 86.0%

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

      3. distribute-frac-neg2 86.0%

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

      4. distribute-neg-frac2 86.0%

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

      5. remove-double-neg 86.0%

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

      6. sqr-neg 86.0%

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

      7. associate-+l+ 86.0%

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

      8. sqr-neg 86.0%

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

      9. distribute-rgt-out 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.55000000000000002e-9 < m < 0.014999999999999999

   1. Initial program 91.7%

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

      1. associate-/l* 91.7%

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

      2. remove-double-neg 91.7%

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

      3. distribute-frac-neg2 91.7%

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

      4. distribute-neg-frac2 91.7%

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

      5. remove-double-neg 91.7%

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

      6. sqr-neg 91.7%

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

      7. associate-+l+ 91.7%

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

      8. sqr-neg 91.7%

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

      9. distribute-rgt-out 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in m around 0 90.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{-9} \lor \neg \left(m \leq 0.015\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.65 \cdot 10^{+199}:\\ \;\;\;\;\frac{\frac{0.001 \cdot \frac{a}{k} - a \cdot 0.01}{k} - a \cdot -0.1}{k}\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{+182} \lor \neg \left(m \leq 1.35 \cdot 10^{+271}\right):\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k} + \left(k + 10\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.65e+199)
   (/ (- (/ (- (* 0.001 (/ a k)) (* a 0.01)) k) (* a -0.1)) k)
   (if (<= m 2.6)
     (/ a (+ (* k (+ k 10.0)) 1.0))
     (if (or (<= m 6e+182) (not (<= m 1.35e+271)))
       (* a (+ (* k (- (* k 99.0) 10.0)) 1.0))
       (/ (/ a (+ (/ 1.0 k) (+ k 10.0))) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.65e+199) {
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	} else if (m <= 2.6) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if ((m <= 6e+182) || !(m <= 1.35e+271)) {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	} else {
		tmp = (a / ((1.0 / k) + (k + 10.0))) / 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 <= (-3.65d+199)) then
        tmp = ((((0.001d0 * (a / k)) - (a * 0.01d0)) / k) - (a * (-0.1d0))) / k
    else if (m <= 2.6d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else if ((m <= 6d+182) .or. (.not. (m <= 1.35d+271))) then
        tmp = a * ((k * ((k * 99.0d0) - 10.0d0)) + 1.0d0)
    else
        tmp = (a / ((1.0d0 / k) + (k + 10.0d0))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.65e+199) {
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	} else if (m <= 2.6) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if ((m <= 6e+182) || !(m <= 1.35e+271)) {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	} else {
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.65e+199:
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k
	elif m <= 2.6:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	elif (m <= 6e+182) or not (m <= 1.35e+271):
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0)
	else:
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.65e+199)
		tmp = Float64(Float64(Float64(Float64(Float64(0.001 * Float64(a / k)) - Float64(a * 0.01)) / k) - Float64(a * -0.1)) / k);
	elseif (m <= 2.6)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif ((m <= 6e+182) || !(m <= 1.35e+271))
		tmp = Float64(a * Float64(Float64(k * Float64(Float64(k * 99.0) - 10.0)) + 1.0));
	else
		tmp = Float64(Float64(a / Float64(Float64(1.0 / k) + Float64(k + 10.0))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.65e+199)
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	elseif (m <= 2.6)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	elseif ((m <= 6e+182) || ~((m <= 1.35e+271)))
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	else
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.65e+199], N[(N[(N[(N[(N[(0.001 * N[(a / k), $MachinePrecision]), $MachinePrecision] - N[(a * 0.01), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] - N[(a * -0.1), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 2.6], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 6e+182], N[Not[LessEqual[m, 1.35e+271]], $MachinePrecision]], N[(a * N[(N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(1.0 / k), $MachinePrecision] + N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -3.6500000000000001e199

   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 m around 0 25.6%

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

   6. Taylor expanded in k around 0 18.4%

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

      1. *-commutative 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in k around -inf 46.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.001 \cdot \frac{a}{k} - 0.01 \cdot a}{k} + -0.1 \cdot a}{k}} \]

   if -3.6500000000000001e199 < m < 2.60000000000000009

   1. Initial program 95.0%

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

      1. associate-/l* 95.0%

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

      2. remove-double-neg 95.0%

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

      3. distribute-frac-neg2 95.0%

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

      4. distribute-neg-frac2 95.0%

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

      5. remove-double-neg 95.0%

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

      6. sqr-neg 95.0%

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

      7. associate-+l+ 95.0%

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

      8. sqr-neg 95.0%

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

      9. distribute-rgt-out 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in m around 0 69.8%

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

   if 2.60000000000000009 < m < 6.0000000000000004e182 or 1.34999999999999995e271 < m

   1. Initial program 66.2%

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

      1. associate-/l* 66.2%

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

      2. remove-double-neg 66.2%

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

      3. distribute-frac-neg2 66.2%

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

      4. distribute-neg-frac2 66.2%

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

      5. remove-double-neg 66.2%

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

      6. sqr-neg 66.2%

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

      7. associate-+l+ 66.2%

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

      8. sqr-neg 66.2%

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

      9. distribute-rgt-out 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 32.7%

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

      1. cancel-sign-sub-inv 32.7%

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

      2. mul-1-neg 32.7%

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

      3. distribute-rgt1-in 32.7%

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

      4. metadata-eval 32.7%

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

      5. metadata-eval 32.7%

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

      6. *-commutative 32.7%

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

   8. Simplified 32.7%

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

   9. Taylor expanded in k around 0 32.7%

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

       1. distribute-lft-in 28.1%

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

       2. *-commutative 28.1%

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

       3. metadata-eval 28.1%

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

       4. distribute-lft-neg-in 28.1%

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

       5. associate-*r* 28.1%

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

       6. *-commutative 28.1%

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

       7. associate-*r* 28.1%

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

       8. distribute-lft-in 32.7%

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

       9. distribute-rgt-neg-in 32.7%

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

       10. distribute-lft-out 32.7%

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

       11. distribute-lft-neg-in 32.7%

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

       12. metadata-eval 32.7%

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

       13. *-commutative 32.7%

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

   11. Simplified 32.7%

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

   12. Taylor expanded in a around 0 42.7%

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

   if 6.0000000000000004e182 < m < 1.34999999999999995e271

   1. Initial program 91.7%

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

      1. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in k around inf 91.7%

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

      1. add-sqr-sqrt 79.2%

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

      2. distribute-lft-out 79.2%

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

      3. times-frac 79.2%

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

      4. associate-+l+ 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. div-inv 79.2%

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

      3. associate-*r* 79.2%

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

      4. add-sqr-sqrt 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in m around 0 39.3%

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

       1. +-commutative 39.3%

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

       2. +-commutative 39.3%

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

       3. associate-+l+ 39.3%

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

   12. Simplified 39.3%

       \[\leadsto \frac{\color{blue}{\frac{a}{\frac{1}{k} + \left(k + 10\right)}}}{k} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.65 \cdot 10^{+199}:\\ \;\;\;\;\frac{\frac{0.001 \cdot \frac{a}{k} - a \cdot 0.01}{k} - a \cdot -0.1}{k}\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{+182} \lor \neg \left(m \leq 1.35 \cdot 10^{+271}\right):\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k} + \left(k + 10\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+180} \lor \neg \left(m \leq 6.5 \cdot 10^{+269}\right):\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k} + \left(k + 10\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.85)
   (/ a (+ (* k (+ k 10.0)) 1.0))
   (if (or (<= m 4.6e+180) (not (<= m 6.5e+269)))
     (* a (+ (* k (- (* k 99.0) 10.0)) 1.0))
     (/ (/ a (+ (/ 1.0 k) (+ k 10.0))) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.85) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if ((m <= 4.6e+180) || !(m <= 6.5e+269)) {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	} else {
		tmp = (a / ((1.0 / k) + (k + 10.0))) / 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 <= 1.85d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else if ((m <= 4.6d+180) .or. (.not. (m <= 6.5d+269))) then
        tmp = a * ((k * ((k * 99.0d0) - 10.0d0)) + 1.0d0)
    else
        tmp = (a / ((1.0d0 / k) + (k + 10.0d0))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.85) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else if ((m <= 4.6e+180) || !(m <= 6.5e+269)) {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	} else {
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.85:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	elif (m <= 4.6e+180) or not (m <= 6.5e+269):
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0)
	else:
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.85)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	elseif ((m <= 4.6e+180) || !(m <= 6.5e+269))
		tmp = Float64(a * Float64(Float64(k * Float64(Float64(k * 99.0) - 10.0)) + 1.0));
	else
		tmp = Float64(Float64(a / Float64(Float64(1.0 / k) + Float64(k + 10.0))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.85)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	elseif ((m <= 4.6e+180) || ~((m <= 6.5e+269)))
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	else
		tmp = (a / ((1.0 / k) + (k + 10.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.85], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 4.6e+180], N[Not[LessEqual[m, 6.5e+269]], $MachinePrecision]], N[(a * N[(N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(1.0 / k), $MachinePrecision] + N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.8500000000000001

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 62.7%

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

   if 1.8500000000000001 < m < 4.5999999999999998e180 or 6.5000000000000003e269 < m

   1. Initial program 66.2%

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

      1. associate-/l* 66.2%

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

      2. remove-double-neg 66.2%

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

      3. distribute-frac-neg2 66.2%

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

      4. distribute-neg-frac2 66.2%

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

      5. remove-double-neg 66.2%

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

      6. sqr-neg 66.2%

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

      7. associate-+l+ 66.2%

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

      8. sqr-neg 66.2%

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

      9. distribute-rgt-out 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 32.7%

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

      1. cancel-sign-sub-inv 32.7%

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

      2. mul-1-neg 32.7%

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

      3. distribute-rgt1-in 32.7%

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

      4. metadata-eval 32.7%

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

      5. metadata-eval 32.7%

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

      6. *-commutative 32.7%

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

   8. Simplified 32.7%

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

   9. Taylor expanded in k around 0 32.7%

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

       1. distribute-lft-in 28.1%

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

       2. *-commutative 28.1%

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

       3. metadata-eval 28.1%

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

       4. distribute-lft-neg-in 28.1%

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

       5. associate-*r* 28.1%

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

       6. *-commutative 28.1%

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

       7. associate-*r* 28.1%

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

       8. distribute-lft-in 32.7%

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

       9. distribute-rgt-neg-in 32.7%

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

       10. distribute-lft-out 32.7%

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

       11. distribute-lft-neg-in 32.7%

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

       12. metadata-eval 32.7%

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

       13. *-commutative 32.7%

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

   11. Simplified 32.7%

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

   12. Taylor expanded in a around 0 42.7%

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

   if 4.5999999999999998e180 < m < 6.5000000000000003e269

   1. Initial program 91.7%

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

      1. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in k around inf 91.7%

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

      1. add-sqr-sqrt 79.2%

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

      2. distribute-lft-out 79.2%

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

      3. times-frac 79.2%

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

      4. associate-+l+ 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. div-inv 79.2%

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

      3. associate-*r* 79.2%

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

      4. add-sqr-sqrt 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in m around 0 39.3%

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

       1. +-commutative 39.3%

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

       2. +-commutative 39.3%

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

       3. associate-+l+ 39.3%

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

   12. Simplified 39.3%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+180} \lor \neg \left(m \leq 6.5 \cdot 10^{+269}\right):\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k} + \left(k + 10\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.85)
   (/ a (+ (* k (+ k 10.0)) 1.0))
   (* a (+ (* k (- (* k 99.0) 10.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.85) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	}
	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 <= 1.85d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = a * ((k * ((k * 99.0d0) - 10.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.85) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.85:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.85)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(Float64(k * 99.0) - 10.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.85)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = a * ((k * ((k * 99.0) - 10.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.85], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.8500000000000001

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 62.7%

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

   if 1.8500000000000001 < m

   1. Initial program 73.0%

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

      1. associate-/l* 73.0%

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

      2. remove-double-neg 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-neg-frac2 73.0%

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

      5. remove-double-neg 73.0%

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

      6. sqr-neg 73.0%

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

      7. associate-+l+ 73.0%

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

      8. sqr-neg 73.0%

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

      9. distribute-rgt-out 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 28.5%

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

      1. cancel-sign-sub-inv 28.5%

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

      2. mul-1-neg 28.5%

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

      3. distribute-rgt1-in 28.5%

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

      4. metadata-eval 28.5%

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

      5. metadata-eval 28.5%

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

      6. *-commutative 28.5%

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

   8. Simplified 28.5%

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

   9. Taylor expanded in k around 0 28.5%

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

       1. distribute-lft-in 22.8%

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

       2. *-commutative 22.8%

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

       3. metadata-eval 22.8%

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

       4. distribute-lft-neg-in 22.8%

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

       5. associate-*r* 22.8%

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

       6. *-commutative 22.8%

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

       7. associate-*r* 22.8%

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

       8. distribute-lft-in 28.5%

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

       9. distribute-rgt-neg-in 28.5%

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

       10. distribute-lft-out 28.5%

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

       11. distribute-lft-neg-in 28.5%

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

       12. metadata-eval 28.5%

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

       13. *-commutative 28.5%

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

   11. Simplified 28.5%

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

   12. Taylor expanded in a around 0 35.8%

       \[\leadsto \color{blue}{a \cdot \left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99 - 10\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-295} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.8e-295) (not (<= k 0.1))) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.8e-295) || !(k <= 0.1)) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 1.8d-295) .or. (.not. (k <= 0.1d0))) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.8e-295) || !(k <= 0.1)) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.8e-295) or not (k <= 0.1):
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.8e-295) || !(k <= 0.1))
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 1.8e-295) || ~((k <= 0.1)))
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.8e-295], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.8000000000000001e-295 or 0.10000000000000001 < k

   1. Initial program 82.0%

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

      1. associate-/l* 82.0%

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

      2. remove-double-neg 82.0%

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

      3. distribute-frac-neg2 82.0%

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

      4. distribute-neg-frac2 82.0%

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

      5. remove-double-neg 82.0%

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

      6. sqr-neg 82.0%

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

      7. associate-+l+ 82.0%

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

      8. sqr-neg 82.0%

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

      9. distribute-rgt-out 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around 0 16.1%

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

      1. *-commutative 16.1%

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

   8. Simplified 16.1%

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

   9. Taylor expanded in k around inf 16.3%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]

   if 1.8000000000000001e-295 < k < 0.10000000000000001

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. remove-double-neg 99.9%

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

      6. sqr-neg 99.9%

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

      7. associate-+l+ 99.9%

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

      8. sqr-neg 99.9%

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

      9. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 49.9%

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

   6. Taylor expanded in k around 0 48.2%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-295} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.5) (/ a (+ (* k 10.0) 1.0)) (+ a (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.5) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = a + (k * (99.0 * (k * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.5d0) then
        tmp = a / ((k * 10.0d0) + 1.0d0)
    else
        tmp = a + (k * (99.0d0 * (k * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.5) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = a + (k * (99.0 * (k * a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.5:
		tmp = a / ((k * 10.0) + 1.0)
	else:
		tmp = a + (k * (99.0 * (k * a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.5)
		tmp = Float64(a / Float64(Float64(k * 10.0) + 1.0));
	else
		tmp = Float64(a + Float64(k * Float64(99.0 * Float64(k * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.5)
		tmp = a / ((k * 10.0) + 1.0);
	else
		tmp = a + (k * (99.0 * (k * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.5], N[(a / N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 62.7%

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

   6. Taylor expanded in k around 0 39.8%

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

      1. *-commutative 39.8%

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

   8. Simplified 39.8%

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

   if 2.5 < m

   1. Initial program 73.0%

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

      1. associate-/l* 73.0%

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

      2. remove-double-neg 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-neg-frac2 73.0%

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

      5. remove-double-neg 73.0%

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

      6. sqr-neg 73.0%

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

      7. associate-+l+ 73.0%

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

      8. sqr-neg 73.0%

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

      9. distribute-rgt-out 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 28.5%

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

      1. cancel-sign-sub-inv 28.5%

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

      2. mul-1-neg 28.5%

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

      3. distribute-rgt1-in 28.5%

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

      4. metadata-eval 28.5%

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

      5. metadata-eval 28.5%

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

      6. *-commutative 28.5%

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

   8. Simplified 28.5%

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

   9. Taylor expanded in k around inf 28.5%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.2) (/ a (+ (* k (+ k 10.0)) 1.0)) (+ a (* k (* 99.0 (* k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a + (k * (99.0 * (k * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.2d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = a + (k * (99.0d0 * (k * a)))
    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 / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a + (k * (99.0 * (k * a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.2:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = a + (k * (99.0 * (k * a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64(a + Float64(k * Float64(99.0 * Float64(k * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = a + (k * (99.0 * (k * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.2], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(99.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 62.7%

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

   if 2.2000000000000002 < m

   1. Initial program 73.0%

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

      1. associate-/l* 73.0%

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

      2. remove-double-neg 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-neg-frac2 73.0%

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

      5. remove-double-neg 73.0%

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

      6. sqr-neg 73.0%

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

      7. associate-+l+ 73.0%

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

      8. sqr-neg 73.0%

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

      9. distribute-rgt-out 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 28.5%

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

      1. cancel-sign-sub-inv 28.5%

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

      2. mul-1-neg 28.5%

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

      3. distribute-rgt1-in 28.5%

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

      4. metadata-eval 28.5%

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

      5. metadata-eval 28.5%

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

      6. *-commutative 28.5%

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

   8. Simplified 28.5%

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

   9. Taylor expanded in k around inf 28.5%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (* a (+ (* k -10.0) 1.0)) (* (/ a k) 0.1)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * ((k * -10.0) + 1.0);
	} else {
		tmp = (a / k) * 0.1;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.075d0) then
        tmp = a * ((k * (-10.0d0)) + 1.0d0)
    else
        tmp = (a / k) * 0.1d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * ((k * -10.0) + 1.0);
	} else {
		tmp = (a / k) * 0.1;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a * ((k * -10.0) + 1.0)
	else:
		tmp = (a / k) * 0.1
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a * Float64(Float64(k * -10.0) + 1.0));
	else
		tmp = Float64(Float64(a / k) * 0.1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a * ((k * -10.0) + 1.0);
	else
		tmp = (a / k) * 0.1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a * N[(N[(k * -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   1. Initial program 91.3%

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

      1. associate-/l* 91.3%

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

      2. remove-double-neg 91.3%

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

      3. distribute-frac-neg2 91.3%

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

      4. distribute-neg-frac2 91.3%

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

      5. remove-double-neg 91.3%

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

      6. sqr-neg 91.3%

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

      7. associate-+l+ 91.3%

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

      8. sqr-neg 91.3%

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

      9. distribute-rgt-out 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in k around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. distribute-lft1-in 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in m around 0 29.6%

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

   if 0.0749999999999999972 < k

   1. Initial program 80.6%

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

      1. associate-/l* 80.6%

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

      2. remove-double-neg 80.6%

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

      3. distribute-frac-neg2 80.6%

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

      4. distribute-neg-frac2 80.6%

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

      5. remove-double-neg 80.6%

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

      6. sqr-neg 80.6%

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

      7. associate-+l+ 80.6%

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

      8. sqr-neg 80.6%

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

      9. distribute-rgt-out 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in m around 0 59.9%

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

   6. Taylor expanded in k around 0 23.0%

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

      1. *-commutative 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in k around inf 23.0%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.2e+23) (/ a (+ (* k 10.0) 1.0)) (* a (+ (* k -10.0) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.2e+23) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = a * ((k * -10.0) + 1.0);
	}
	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 <= 3.2d+23) then
        tmp = a / ((k * 10.0d0) + 1.0d0)
    else
        tmp = a * ((k * (-10.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.2e+23) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = a * ((k * -10.0) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 3.2e+23:
		tmp = a / ((k * 10.0) + 1.0)
	else:
		tmp = a * ((k * -10.0) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.2e+23)
		tmp = Float64(a / Float64(Float64(k * 10.0) + 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * -10.0) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 3.2e+23)
		tmp = a / ((k * 10.0) + 1.0);
	else
		tmp = a * ((k * -10.0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.2e+23], N[(a / N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.2e23

   1. Initial program 95.4%

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

      1. associate-/l* 95.4%

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

      2. remove-double-neg 95.4%

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

      3. distribute-frac-neg2 95.4%

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

      4. distribute-neg-frac2 95.4%

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

      5. remove-double-neg 95.4%

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

      6. sqr-neg 95.4%

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

      7. associate-+l+ 95.4%

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

      8. sqr-neg 95.4%

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

      9. distribute-rgt-out 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in m around 0 61.0%

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

   6. Taylor expanded in k around 0 38.7%

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

      1. *-commutative 38.7%

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

   8. Simplified 38.7%

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

   if 3.2e23 < m

   1. Initial program 72.6%

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

      1. associate-/l* 72.6%

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

      2. remove-double-neg 72.6%

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

      3. distribute-frac-neg2 72.6%

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

      4. distribute-neg-frac2 72.6%

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

      5. remove-double-neg 72.6%

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

      6. sqr-neg 72.6%

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

      7. associate-+l+ 72.6%

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

      8. sqr-neg 72.6%

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

      9. distribute-rgt-out 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in k around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. distribute-lft1-in 79.8%

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

      3. *-commutative 79.8%

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

      4. fma-define 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in m around 0 11.6%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.5% 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 87.9%

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

   1. associate-/l* 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-frac-neg2 87.9%

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

   4. distribute-neg-frac2 87.9%

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

   5. remove-double-neg 87.9%

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

   6. sqr-neg 87.9%

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

   7. associate-+l+ 87.9%

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

   8. sqr-neg 87.9%

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

   9. distribute-rgt-out 87.9%

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

3. Simplified 87.9%

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

5. Taylor expanded in m around 0 42.0%

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

6. Taylor expanded in k around 0 18.8%

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

7. Final simplification 18.8%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))