Falkner and Boettcher, Appendix A

Percentage Accurate: 90.7% → 97.6%

Time: 9.5s

Alternatives: 14

Speedup: 1.1×

• 22.9% 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: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{+160}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2.6e+160)
   (* a (pow k m))
   (if (<= k 1.05e+18)
     (* a (/ (pow k m) (+ (* k (+ k 10.0)) 1.0)))
     (/ (/ (pow k m) k) (/ k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.6e+160) {
		tmp = a * pow(k, m);
	} else if (k <= 1.05e+18) {
		tmp = a * (pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = (pow(k, m) / k) / (k / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-2.6d+160)) then
        tmp = a * (k ** m)
    else if (k <= 1.05d+18) then
        tmp = a * ((k ** m) / ((k * (k + 10.0d0)) + 1.0d0))
    else
        tmp = ((k ** m) / k) / (k / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.6e+160) {
		tmp = a * Math.pow(k, m);
	} else if (k <= 1.05e+18) {
		tmp = a * (Math.pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = (Math.pow(k, m) / k) / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2.6e+160:
		tmp = a * math.pow(k, m)
	elif k <= 1.05e+18:
		tmp = a * (math.pow(k, m) / ((k * (k + 10.0)) + 1.0))
	else:
		tmp = (math.pow(k, m) / k) / (k / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2.6e+160)
		tmp = Float64(a * (k ^ m));
	elseif (k <= 1.05e+18)
		tmp = Float64(a * Float64((k ^ m) / Float64(Float64(k * Float64(k + 10.0)) + 1.0)));
	else
		tmp = Float64(Float64((k ^ m) / k) / Float64(k / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2.6e+160)
		tmp = a * (k ^ m);
	elseif (k <= 1.05e+18)
		tmp = a * ((k ^ m) / ((k * (k + 10.0)) + 1.0));
	else
		tmp = ((k ^ m) / k) / (k / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2.6e+160], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.05e+18], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / N[(k / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.6e160

   1. Initial program 33.3%

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

      1. associate-*r/ 33.3%

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

      2. associate-+l+ 33.3%

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

      3. +-commutative 33.3%

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

      4. distribute-rgt-out 33.3%

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

      5. fma-def 33.3%

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

      6. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in k around 0 0.0%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if -2.6e160 < k < 1.05e18

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 1.05e18 < k

   1. Initial program 81.7%

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

      1. associate-*r/ 81.7%

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

      2. associate-+l+ 81.7%

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

      3. +-commutative 81.7%

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

      4. distribute-rgt-out 81.7%

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

      5. fma-def 81.7%

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

      6. +-commutative 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in k around inf 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. associate-/l* 79.1%

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

      4. mul-1-neg 79.1%

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

      5. distribute-rgt-neg-in 79.1%

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

      6. log-rec 79.1%

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

      7. associate-/l* 88.5%

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

   6. Simplified 88.5%

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

      1. add-sqr-sqrt 88.5%

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

      2. associate-/r/ 88.5%

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

      3. times-frac 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. associate-*r/ 98.6%

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

      3. add-sqr-sqrt 98.6%

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

   10. Applied egg-rr 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{+160}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 2: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. distribute-rgt-out 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in k around 0 53.4%

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

      1. exp-to-pow 99.6%

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

   6. Simplified 99.6%

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

   if 1 < k

   1. Initial program 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in k around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. associate-/l* 76.5%

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

      4. mul-1-neg 76.5%

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

      5. distribute-rgt-neg-in 76.5%

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

      6. log-rec 76.5%

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

      7. associate-/l* 85.1%

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

   6. Simplified 85.1%

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

      1. add-sqr-sqrt 85.1%

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

      2. associate-/r/ 85.1%

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

      3. times-frac 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. associate-*l/ 94.4%

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

      2. associate-*r/ 94.4%

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

      3. add-sqr-sqrt 94.4%

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

   10. Applied egg-rr 94.4%

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

4. Final simplification 97.8%

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

Alternative 3: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.6 \cdot 10^{-7} \lor \neg \left(m \leq 3 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -7.6e-7) (not (<= m 3e-16)))
   (* a (pow k m))
   (/ a (+ (* k (+ k 10.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -7.6e-7) || !(m <= 3e-16)) {
		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 <= (-7.6d-7)) .or. (.not. (m <= 3d-16))) 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 <= -7.6e-7) || !(m <= 3e-16)) {
		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 <= -7.6e-7) or not (m <= 3e-16):
		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 <= -7.6e-7) || !(m <= 3e-16))
		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 <= -7.6e-7) || ~((m <= 3e-16)))
		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, -7.6e-7], N[Not[LessEqual[m, 3e-16]], $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 < -7.60000000000000029e-7 or 2.99999999999999994e-16 < m

   1. Initial program 88.2%

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

      1. associate-*r/ 88.2%

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

      2. associate-+l+ 88.2%

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

      3. +-commutative 88.2%

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

      4. distribute-rgt-out 88.2%

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

      5. fma-def 88.2%

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

      6. +-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in k around 0 47.9%

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

      1. exp-to-pow 98.9%

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

   6. Simplified 98.9%

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

   if -7.60000000000000029e-7 < m < 2.99999999999999994e-16

   1. Initial program 93.5%

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

      1. associate-*r/ 93.5%

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

      2. associate-+l+ 93.5%

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

      3. +-commutative 93.5%

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

      4. distribute-rgt-out 93.5%

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

      5. fma-def 93.5%

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

      6. +-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in m around 0 93.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.6 \cdot 10^{-7} \lor \neg \left(m \leq 3 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \end{array} \]

Alternative 4: 61.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.7e+20)
   (* a (/ 1.0 (* k k)))
   (if (<= m 4.2e-5)
     (/ a (+ (* k (+ k 10.0)) 1.0))
     (+ a (* a (* k (- (* k 100.0) 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4.2e-5) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a + (a * (k * ((k * 100.0) - 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.7d+20)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 4.2d-5) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = a + (a * (k * ((k * 100.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4.2e-5) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a + (a * (k * ((k * 100.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.7e+20:
		tmp = a * (1.0 / (k * k))
	elif m <= 4.2e-5:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = a + (a * (k * ((k * 100.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.7e+20)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 4.2e-5)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(Float64(k * 100.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.7e+20)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 4.2e-5)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = a + (a * (k * ((k * 100.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.7e+20], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-5], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(N[(k * 100.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.7e20

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.9%

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

   5. Taylor expanded in k around inf 61.2%

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

      1. unpow2 61.2%

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

   7. Simplified 61.2%

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

   if -3.7e20 < m < 4.19999999999999977e-5

   1. Initial program 94.0%

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

      1. associate-*r/ 94.0%

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

      2. associate-+l+ 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt-out 94.0%

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

      5. fma-def 94.0%

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

      6. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in m around 0 88.5%

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

   if 4.19999999999999977e-5 < m

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. associate-+l+ 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-rgt-out 79.5%

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

      5. fma-def 79.5%

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

      6. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 3.1%

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

      1. *-commutative 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in k around 0 26.8%

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

      1. +-commutative 26.8%

         \[\leadsto a + \color{blue}{\left(100 \cdot \left({k}^{2} \cdot a\right) + -10 \cdot \left(k \cdot a\right)\right)} \]

      2. unpow2 26.8%

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

      3. associate-*r* 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*r* 26.8%

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

      6. metadata-eval 26.8%

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

      7. distribute-lft-neg-in 26.8%

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

      8. *-commutative 26.8%

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

      9. distribute-rgt-out 31.3%

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

      10. sub-neg 31.3%

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

      11. associate-*l* 31.3%

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

      12. distribute-lft-out-- 32.4%

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

   10. Simplified 32.4%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot \left(k \cdot 100 - 10\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 - 10\right)\right)\\ \end{array} \]

Alternative 5: 46.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-171}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -8.5e-171)
   (* a (/ 1.0 (* k k)))
   (if (<= k 1.1e-308)
     (* -10.0 (* k a))
     (if (<= k 1.0) a (/ 1.0 (* k (/ k a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -8.5e-171) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 1.1e-308) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-8.5d-171)) then
        tmp = a * (1.0d0 / (k * k))
    else if (k <= 1.1d-308) then
        tmp = (-10.0d0) * (k * a)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -8.5e-171) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 1.1e-308) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -8.5e-171:
		tmp = a * (1.0 / (k * k))
	elif k <= 1.1e-308:
		tmp = -10.0 * (k * a)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -8.5e-171)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (k <= 1.1e-308)
		tmp = Float64(-10.0 * Float64(k * a));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -8.5e-171)
		tmp = a * (1.0 / (k * k));
	elseif (k <= 1.1e-308)
		tmp = -10.0 * (k * a);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -8.5e-171], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e-308], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < -8.50000000000000032e-171

   1. Initial program 84.4%

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

      1. associate-*r/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. distribute-rgt-out 84.4%

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

      5. fma-def 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in m around 0 15.2%

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

   5. Taylor expanded in k around inf 19.5%

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

      1. unpow2 19.5%

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

   7. Simplified 19.5%

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

   if -8.50000000000000032e-171 < k < 1.1000000000000001e-308

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.3%

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

   5. Taylor expanded in k around 0 4.3%

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

   6. Taylor expanded in k around inf 51.8%

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

   if 1.1000000000000001e-308 < k < 1

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 63.4%

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

   5. Taylor expanded in k around 0 63.4%

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

   if 1 < k

   1. Initial program 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 64.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

      1. un-div-inv 60.6%

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

      2. clear-num 60.8%

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

      3. associate-*l/ 65.1%

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

      4. *-commutative 65.1%

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

   9. Applied egg-rr 65.1%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-171}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 6: 47.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{-172}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2.1e-172)
   (* a (/ 1.0 (* k k)))
   (if (<= k 1.1e-308)
     (* -10.0 (* k a))
     (if (<= k 10.2) (/ a (+ 1.0 (* k 10.0))) (/ 1.0 (* k (/ k a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.1e-172) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 1.1e-308) {
		tmp = -10.0 * (k * a);
	} else if (k <= 10.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-2.1d-172)) then
        tmp = a * (1.0d0 / (k * k))
    else if (k <= 1.1d-308) then
        tmp = (-10.0d0) * (k * a)
    else if (k <= 10.2d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.1e-172) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= 1.1e-308) {
		tmp = -10.0 * (k * a);
	} else if (k <= 10.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2.1e-172:
		tmp = a * (1.0 / (k * k))
	elif k <= 1.1e-308:
		tmp = -10.0 * (k * a)
	elif k <= 10.2:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2.1e-172)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (k <= 1.1e-308)
		tmp = Float64(-10.0 * Float64(k * a));
	elseif (k <= 10.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2.1e-172)
		tmp = a * (1.0 / (k * k));
	elseif (k <= 1.1e-308)
		tmp = -10.0 * (k * a);
	elseif (k <= 10.2)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2.1e-172], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e-308], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.2], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < -2.0999999999999999e-172

   1. Initial program 84.4%

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

      1. associate-*r/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. distribute-rgt-out 84.4%

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

      5. fma-def 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in m around 0 15.2%

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

   5. Taylor expanded in k around inf 19.5%

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

      1. unpow2 19.5%

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

   7. Simplified 19.5%

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

   if -2.0999999999999999e-172 < k < 1.1000000000000001e-308

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.3%

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

   5. Taylor expanded in k around 0 4.3%

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

   6. Taylor expanded in k around inf 51.8%

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

   if 1.1000000000000001e-308 < k < 10.199999999999999

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 63.8%

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

   5. Taylor expanded in k around 0 62.9%

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

      1. *-commutative 62.9%

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

   7. Simplified 62.9%

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

   if 10.199999999999999 < k

   1. Initial program 83.0%

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

      1. associate-*r/ 83.0%

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

      2. associate-+l+ 83.0%

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

      3. +-commutative 83.0%

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

      4. distribute-rgt-out 83.0%

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

      5. fma-def 83.0%

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

      6. +-commutative 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in m around 0 64.5%

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

   5. Taylor expanded in k around inf 61.1%

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

      1. unpow2 61.1%

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

   7. Simplified 61.1%

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

      1. un-div-inv 61.1%

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

      2. clear-num 61.3%

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

      3. associate-*l/ 65.7%

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

      4. *-commutative 65.7%

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

   9. Applied egg-rr 65.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{-172}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 7: 58.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 980:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.7e+20)
   (* a (/ 1.0 (* k k)))
   (if (<= m 980.0) (/ a (+ (* k (+ k 10.0)) 1.0)) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 980.0) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = -10.0 * (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 <= (-3.7d+20)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 980.0d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 980.0) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.7e+20:
		tmp = a * (1.0 / (k * k))
	elif m <= 980.0:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.7e+20)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 980.0)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.7e+20)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 980.0)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.7e+20], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 980.0], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.7e20

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.9%

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

   5. Taylor expanded in k around inf 61.2%

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

      1. unpow2 61.2%

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

   7. Simplified 61.2%

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

   if -3.7e20 < m < 980

   1. Initial program 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. distribute-rgt-out 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 87.2%

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

   if 980 < m

   1. Initial program 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-+l+ 79.1%

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

      3. +-commutative 79.1%

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

      4. distribute-rgt-out 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 10.0%

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

   6. Taylor expanded in k around inf 24.1%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 980:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8: 46.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.15 \cdot 10^{-173}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -3.15e-173)
   (* a (/ 1.0 (* k k)))
   (if (<= k -5e-311) (* -10.0 (* k a)) (if (<= k 1.0) a (/ (/ a k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.15e-173) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / 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 <= (-3.15d-173)) then
        tmp = a * (1.0d0 / (k * k))
    else if (k <= (-5d-311)) then
        tmp = (-10.0d0) * (k * a)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.15e-173) {
		tmp = a * (1.0 / (k * k));
	} else if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -3.15e-173:
		tmp = a * (1.0 / (k * k))
	elif k <= -5e-311:
		tmp = -10.0 * (k * a)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -3.15e-173)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (k <= -5e-311)
		tmp = Float64(-10.0 * Float64(k * a));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -3.15e-173)
		tmp = a * (1.0 / (k * k));
	elseif (k <= -5e-311)
		tmp = -10.0 * (k * a);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -3.15e-173], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -5e-311], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < -3.14999999999999984e-173

   1. Initial program 84.4%

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

      1. associate-*r/ 84.4%

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

      2. associate-+l+ 84.4%

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

      3. +-commutative 84.4%

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

      4. distribute-rgt-out 84.4%

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

      5. fma-def 84.4%

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

      6. +-commutative 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in m around 0 15.2%

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

   5. Taylor expanded in k around inf 19.5%

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

      1. unpow2 19.5%

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

   7. Simplified 19.5%

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

   if -3.14999999999999984e-173 < k < -5.00000000000023e-311

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.3%

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

   5. Taylor expanded in k around 0 4.3%

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

   6. Taylor expanded in k around inf 51.8%

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

   if -5.00000000000023e-311 < k < 1

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 63.4%

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

   5. Taylor expanded in k around 0 63.4%

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

   if 1 < k

   1. Initial program 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 64.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

      1. un-div-inv 60.6%

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

      2. associate-/r* 64.9%

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

   9. Applied egg-rr 64.9%

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.15 \cdot 10^{-173}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9: 57.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4200000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.7e+20)
   (* a (/ 1.0 (* k k)))
   (if (<= m 4200000.0) (/ a (+ 1.0 (* k k))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4200000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (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 <= (-3.7d+20)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 4200000.0d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.7e+20) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 4200000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.7e+20:
		tmp = a * (1.0 / (k * k))
	elif m <= 4200000.0:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.7e+20)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 4200000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.7e+20)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 4200000.0)
		tmp = a / (1.0 + (k * k));
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.7e+20], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4200000.0], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.7e20

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.9%

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

   5. Taylor expanded in k around inf 61.2%

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

      1. unpow2 61.2%

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

   7. Simplified 61.2%

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

   if -3.7e20 < m < 4.2e6

   1. Initial program 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. distribute-rgt-out 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 87.2%

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

   5. Taylor expanded in k around inf 83.9%

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

      1. unpow2 83.9%

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

   7. Simplified 83.9%

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

   if 4.2e6 < m

   1. Initial program 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-+l+ 79.1%

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

      3. +-commutative 79.1%

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

      4. distribute-rgt-out 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 10.0%

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

   6. Taylor expanded in k around inf 24.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 4200000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10: 30.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 1000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.4e-44) (/ a (* k 10.0)) (if (<= m 1000.0) a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.4e-44) {
		tmp = a / (k * 10.0);
	} else if (m <= 1000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (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 <= (-3.4d-44)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 1000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.4e-44) {
		tmp = a / (k * 10.0);
	} else if (m <= 1000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.4e-44:
		tmp = a / (k * 10.0)
	elif m <= 1000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.4e-44)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 1000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.4e-44)
		tmp = a / (k * 10.0);
	elseif (m <= 1000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.4e-44], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.40000000000000016e-44

   1. Initial program 97.4%

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

      1. associate-*r/ 97.4%

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

      2. associate-+l+ 97.4%

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

      3. +-commutative 97.4%

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

      4. distribute-rgt-out 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in m around 0 40.1%

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

   5. Taylor expanded in k around 0 15.8%

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

      1. *-commutative 15.8%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in k around inf 20.3%

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

      1. *-commutative 20.3%

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

   10. Simplified 20.3%

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

   if -3.40000000000000016e-44 < m < 1e3

   1. Initial program 95.2%

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

      1. associate-*r/ 95.2%

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

      2. associate-+l+ 95.2%

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

      3. +-commutative 95.2%

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

      4. distribute-rgt-out 95.2%

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

      5. fma-def 95.2%

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

      6. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in m around 0 92.2%

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

   5. Taylor expanded in k around 0 56.5%

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

   if 1e3 < m

   1. Initial program 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-+l+ 79.1%

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

      3. +-commutative 79.1%

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

      4. distribute-rgt-out 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 10.0%

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

   6. Taylor expanded in k around inf 24.1%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 1000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11: 42.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -5e-311) (* -10.0 (* k a)) (if (<= k 1.0) a (/ a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = a / (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 <= (-5d-311)) then
        tmp = (-10.0d0) * (k * a)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = a / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -5e-311:
		tmp = -10.0 * (k * a)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = a / (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -5e-311)
		tmp = Float64(-10.0 * Float64(k * a));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -5e-311)
		tmp = -10.0 * (k * a);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = a / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -5e-311], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -5.00000000000023e-311

   1. Initial program 87.2%

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

      1. associate-*r/ 87.2%

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

      2. associate-+l+ 87.2%

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

      3. +-commutative 87.2%

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

      4. distribute-rgt-out 87.2%

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

      5. fma-def 87.2%

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

      6. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in m around 0 13.2%

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

   5. Taylor expanded in k around 0 11.0%

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

   6. Taylor expanded in k around inf 20.1%

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

   if -5.00000000000023e-311 < k < 1

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 63.4%

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

   5. Taylor expanded in k around 0 63.4%

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

   if 1 < k

   1. Initial program 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 64.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 12: 43.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -5e-311) (* -10.0 (* k a)) (if (<= k 1.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / 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 <= (-5d-311)) then
        tmp = (-10.0d0) * (k * a)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-311) {
		tmp = -10.0 * (k * a);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -5e-311:
		tmp = -10.0 * (k * a)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -5e-311)
		tmp = Float64(-10.0 * Float64(k * a));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -5e-311)
		tmp = -10.0 * (k * a);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -5e-311], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -5.00000000000023e-311

   1. Initial program 87.2%

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

      1. associate-*r/ 87.2%

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

      2. associate-+l+ 87.2%

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

      3. +-commutative 87.2%

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

      4. distribute-rgt-out 87.2%

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

      5. fma-def 87.2%

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

      6. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in m around 0 13.2%

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

   5. Taylor expanded in k around 0 11.0%

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

   6. Taylor expanded in k around inf 20.1%

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

   if -5.00000000000023e-311 < k < 1

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 63.4%

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

   5. Taylor expanded in k around 0 63.4%

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

   if 1 < k

   1. Initial program 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-+l+ 83.2%

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

      3. +-commutative 83.2%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 64.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

      1. un-div-inv 60.6%

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

      2. associate-/r* 64.9%

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

   9. Applied egg-rr 64.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13: 25.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1300:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1300.0) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1300.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (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 <= 1300.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1300.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1300.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1300.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1300.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1300.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1300

   1. Initial program 96.1%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 71.7%

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

   5. Taylor expanded in k around 0 36.0%

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

   if 1300 < m

   1. Initial program 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-+l+ 79.1%

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

      3. +-commutative 79.1%

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

      4. distribute-rgt-out 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 10.0%

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

   6. Taylor expanded in k around inf 24.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1300:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 14: 20.0% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.4%

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

   1. associate-*r/ 90.4%

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

   2. associate-+l+ 90.4%

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

   3. +-commutative 90.4%

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

   4. distribute-rgt-out 90.4%

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

   5. fma-def 90.4%

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

   6. +-commutative 90.4%

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

3. Simplified 90.4%

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

4. Taylor expanded in m around 0 48.6%

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

5. Taylor expanded in k around 0 25.2%

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

6. Final simplification 25.2%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023196 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))