Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.1%

Time: 8.7s

Alternatives: 14

Speedup: 1.1×

• 22.7% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(k, -10, 1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 0.1) (* (fma k -10.0 1.0) t_0) (/ (/ t_0 k) k))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 0.1) {
		tmp = fma(k, -10.0, 1.0) * t_0;
	} else {
		tmp = (t_0 / k) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 0.1)
		tmp = Float64(fma(k, -10.0, 1.0) * t_0);
	else
		tmp = Float64(Float64(t_0 / k) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.1], N[(N[(k * -10.0 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 93.9%

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

      1. associate-*r/ 93.9%

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

      2. associate-+l+ 93.9%

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

      3. +-commutative 93.9%

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

      4. distribute-rgt-out 93.9%

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

      5. fma-def 93.9%

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

      6. +-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in k around 0 33.0%

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

      1. exp-to-pow 33.0%

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

      2. exp-to-pow 33.0%

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

      3. exp-to-pow 33.0%

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

      4. associate-*r* 33.0%

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

      5. exp-to-pow 85.4%

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

      6. distribute-lft1-in 99.5%

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

      7. *-commutative 99.5%

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

      8. fma-def 99.5%

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

      9. *-commutative 99.5%

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

   6. Simplified 99.5%

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

   if 0.10000000000000001 < k

   1. Initial program 85.3%

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

   2. Taylor expanded in k around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. *-commutative 84.4%

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

      2. add-cube-cbrt 84.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{k \cdot k} \]

      3. times-frac 98.7%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k}} \]

      4. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      5. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      6. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}}{k} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}} \]
   7. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}}{k}} \]

      2. associate-*r/ 98.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}}{k} \]

      3. unpow2 98.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right)} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}{k} \]

      4. add-cube-cbrt 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(k, -10, 1\right) \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.9%

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

      1. associate-*r/ 93.9%

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

      2. associate-+l+ 93.9%

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

      3. +-commutative 93.9%

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

      4. distribute-rgt-out 93.9%

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

      5. fma-def 93.9%

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

      6. +-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in k around 0 46.7%

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

      1. exp-to-pow 99.1%

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

   6. Simplified 99.1%

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

   if 1 < k

   1. Initial program 85.3%

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

   2. Taylor expanded in k around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. *-commutative 84.4%

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

      2. add-cube-cbrt 84.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{k \cdot k} \]

      3. times-frac 98.7%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k}} \]

      4. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      5. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      6. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}}{k} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}} \]
   7. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}}{k}} \]

      2. associate-*r/ 98.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}}{k} \]

      3. unpow2 98.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right)} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}{k} \]

      4. add-cube-cbrt 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

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

Alternative 3: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-6} \lor \neg \left(m \leq 1.32\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.8e-6) (not (<= m 1.32)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.8e-6) || !(m <= 1.32)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-5.8d-6)) .or. (.not. (m <= 1.32d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.8e-6) || !(m <= 1.32)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -5.8e-6) or not (m <= 1.32):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5.8e-6) || !(m <= 1.32))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -5.8e-6) || ~((m <= 1.32)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5.8e-6], N[Not[LessEqual[m, 1.32]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.8000000000000004e-6 or 1.32000000000000006 < m

   1. Initial program 87.3%

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

      1. associate-*r/ 87.3%

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

      2. associate-+l+ 87.3%

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

      3. +-commutative 87.3%

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

      4. distribute-rgt-out 87.3%

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

      5. fma-def 87.3%

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

      6. +-commutative 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in k around 0 53.0%

      \[\leadsto a \cdot \color{blue}{e^{\log k \cdot m}} \]
   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 -5.8000000000000004e-6 < m < 1.32000000000000006

   1. Initial program 95.8%

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

      1. associate-*r/ 95.8%

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

      2. associate-+l+ 95.8%

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

      3. +-commutative 95.8%

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

      4. distribute-rgt-out 95.8%

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

      5. fma-def 95.8%

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

      6. +-commutative 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in m around 0 95.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-6} \lor \neg \left(m \leq 1.32\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* a (pow k m)) (/ a (pow k (- 2.0 m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / pow(k, (2.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = a / (k ** (2.0d0 - m))
    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 = a / Math.pow(k, (2.0 - m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = a / math.pow(k, (2.0 - m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / (k ^ Float64(2.0 - m)));
	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 = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, N[(2.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 93.9%

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

      1. associate-*r/ 93.9%

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

      2. associate-+l+ 93.9%

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

      3. +-commutative 93.9%

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

      4. distribute-rgt-out 93.9%

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

      5. fma-def 93.9%

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

      6. +-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in k around 0 46.7%

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

      1. exp-to-pow 99.1%

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

   6. Simplified 99.1%

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

   if 1 < k

   1. Initial program 85.3%

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

   2. Taylor expanded in k around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. expm1-log1p-u 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{k \cdot k}\right)\right)} \]

      2. expm1-udef 63.2%

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

      3. associate-/l* 63.2%

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

      4. pow2 63.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      5. pow-div 69.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 83.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 95.6%

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

   8. Simplified 95.6%

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

4. Final simplification 97.6%

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

Alternative 5: 58.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;a \cdot \frac{--1}{k \cdot k}\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.3)
   (* a (/ (- -1.0) (* k k)))
   (if (<= m 7.2e+15) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.3) {
		tmp = a * (-(-1.0) / (k * k));
	} else if (m <= 7.2e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} 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 <= (-0.3d0)) then
        tmp = a * (-(-1.0d0) / (k * k))
    else if (m <= 7.2d+15) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    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 <= -0.3) {
		tmp = a * (-(-1.0) / (k * k));
	} else if (m <= 7.2e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.3:
		tmp = a * (-(-1.0) / (k * k))
	elif m <= 7.2e+15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.3)
		tmp = Float64(a * Float64(Float64(-(-1.0)) / Float64(k * k)));
	elseif (m <= 7.2e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 <= -0.3)
		tmp = a * (-(-1.0) / (k * k));
	elseif (m <= 7.2e+15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.3], N[(a * N[((--1.0) / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.2e+15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.299999999999999989

   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 38.4%

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

   5. Taylor expanded in k around inf 64.7%

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

      1. unpow2 64.7%

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

   7. Simplified 64.7%

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

      1. frac-2neg 64.7%

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

      2. div-inv 65.9%

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

      3. distribute-rgt-neg-in 65.9%

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

   9. Applied egg-rr 65.9%

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

   10. Taylor expanded in k around 0 65.9%

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

       1. unpow2 65.9%

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

   12. Simplified 65.9%

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

   if -0.299999999999999989 < m < 7.2e15

   1. Initial program 96.0%

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

      1. associate-*r/ 95.9%

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

      2. associate-+l+ 95.9%

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

      3. +-commutative 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. fma-def 95.9%

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

      6. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in m around 0 91.6%

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

   if 7.2e15 < m

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. associate-+l+ 73.1%

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

      3. +-commutative 73.1%

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

      4. distribute-rgt-out 73.1%

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

      5. fma-def 73.1%

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

      6. +-commutative 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in m around 0 2.5%

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

   5. Taylor expanded in k around 0 9.8%

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

   6. Taylor expanded in k around inf 17.5%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;a \cdot \frac{--1}{k \cdot k}\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 6: 47.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5.3e-287)
   (/ a (* k k))
   (if (<= k 0.1) (* a (+ 1.0 (* k -10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5.3e-287) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} 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 <= 5.3d-287) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    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 <= 5.3e-287) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 5.3e-287:
		tmp = a / (k * k)
	elif k <= 0.1:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5.3e-287)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 5.3e-287)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5.3e-287], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.29999999999999949e-287

   1. Initial program 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-+l+ 88.8%

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

      3. +-commutative 88.8%

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

      4. distribute-rgt-out 88.8%

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

      5. fma-def 88.8%

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

      6. +-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in m around 0 20.2%

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

   5. Taylor expanded in k around inf 34.3%

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

      1. unpow2 34.3%

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

   7. Simplified 34.3%

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

   if 5.29999999999999949e-287 < k < 0.10000000000000001

   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 k around 0 71.3%

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

      1. exp-to-pow 71.3%

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

      2. exp-to-pow 71.3%

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

      3. exp-to-pow 71.3%

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

      4. associate-*r* 71.3%

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

      5. exp-to-pow 71.3%

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

      6. distribute-lft1-in 98.9%

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

      7. *-commutative 98.9%

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

      8. fma-def 98.9%

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

      9. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around 0 54.8%

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

   if 0.10000000000000001 < k

   1. Initial program 85.3%

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

   2. Taylor expanded in k around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. *-commutative 84.4%

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

      2. add-cube-cbrt 84.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{k \cdot k} \]

      3. times-frac 98.7%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k}} \]

      4. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      5. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      6. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}}{k} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}} \]
   7. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}}{k}} \]

      2. associate-*r/ 98.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}}{k} \]

      3. unpow2 98.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right)} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}{k} \]

      4. add-cube-cbrt 99.0%

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

   8. Applied egg-rr 99.0%

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

   9. Taylor expanded in m around 0 60.7%

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

       1. unpow2 60.7%

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

       2. associate-/r* 61.1%

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

   11. Simplified 61.1%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7: 47.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-286}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.4e-286)
   (/ a (* k k))
   (if (<= k 0.1) (* a (+ 1.0 (* k -10.0))) (/ 1.0 (* k (/ k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.4e-286) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		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 <= 2.4d-286) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) 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 <= 2.4e-286) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		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 <= 2.4e-286:
		tmp = a / (k * k)
	elif k <= 0.1:
		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 <= 2.4e-286)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		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 <= 2.4e-286)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		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, 2.4e-286], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], 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 3 regimes

2. if k < 2.39999999999999993e-286

   1. Initial program 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-+l+ 88.8%

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

      3. +-commutative 88.8%

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

      4. distribute-rgt-out 88.8%

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

      5. fma-def 88.8%

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

      6. +-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in m around 0 20.2%

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

   5. Taylor expanded in k around inf 34.3%

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

      1. unpow2 34.3%

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

   7. Simplified 34.3%

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

   if 2.39999999999999993e-286 < k < 0.10000000000000001

   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 k around 0 71.3%

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

      1. exp-to-pow 71.3%

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

      2. exp-to-pow 71.3%

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

      3. exp-to-pow 71.3%

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

      4. associate-*r* 71.3%

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

      5. exp-to-pow 71.3%

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

      6. distribute-lft1-in 98.9%

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

      7. *-commutative 98.9%

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

      8. fma-def 98.9%

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

      9. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around 0 54.8%

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

   if 0.10000000000000001 < k

   1. Initial program 85.3%

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

      1. associate-*r/ 85.2%

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

      2. associate-+l+ 85.2%

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

      3. +-commutative 85.2%

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

      4. distribute-rgt-out 85.2%

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

      5. fma-def 85.2%

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

      6. +-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in m around 0 61.3%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. unpow2 60.7%

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

   7. Simplified 60.7%

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

      1. associate-/r* 61.1%

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

      2. div-inv 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. clear-num 61.0%

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

       2. frac-times 61.4%

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

       3. metadata-eval 61.4%

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-286}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 8: 47.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5.5e-288)
   (/ a (* k k))
   (if (<= k 10.0) (/ a (+ 1.0 (* k 10.0))) (/ 1.0 (* k (/ k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5.5e-288) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		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 <= 5.5d-288) then
        tmp = a / (k * k)
    else if (k <= 10.0d0) 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 <= 5.5e-288) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		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 <= 5.5e-288:
		tmp = a / (k * k)
	elif k <= 10.0:
		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 <= 5.5e-288)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 10.0)
		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 <= 5.5e-288)
		tmp = a / (k * k);
	elseif (k <= 10.0)
		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, 5.5e-288], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.0], 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 3 regimes

2. if k < 5.5e-288

   1. Initial program 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-+l+ 88.8%

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

      3. +-commutative 88.8%

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

      4. distribute-rgt-out 88.8%

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

      5. fma-def 88.8%

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

      6. +-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in m around 0 20.2%

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

   5. Taylor expanded in k around inf 34.3%

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

      1. unpow2 34.3%

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

   7. Simplified 34.3%

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

   if 5.5e-288 < k < 10

   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 55.9%

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

   5. Taylor expanded in k around 0 54.9%

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

      1. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   if 10 < k

   1. Initial program 85.3%

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

      1. associate-*r/ 85.2%

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

      2. associate-+l+ 85.2%

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

      3. +-commutative 85.2%

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

      4. distribute-rgt-out 85.2%

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

      5. fma-def 85.2%

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

      6. +-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in m around 0 61.3%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. unpow2 60.7%

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

   7. Simplified 60.7%

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

      1. associate-/r* 61.1%

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

      2. div-inv 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. clear-num 61.0%

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

       2. frac-times 61.4%

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

       3. metadata-eval 61.4%

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 9: 57.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.25)
   (/ a (* k k))
   (if (<= m 7.5e+15) (/ a (+ 1.0 (* k k))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.25) {
		tmp = a / (k * k);
	} else if (m <= 7.5e+15) {
		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 <= (-0.25d0)) then
        tmp = a / (k * k)
    else if (m <= 7.5d+15) 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 <= -0.25) {
		tmp = a / (k * k);
	} else if (m <= 7.5e+15) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.25:
		tmp = a / (k * k)
	elif m <= 7.5e+15:
		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 <= -0.25)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 7.5e+15)
		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 <= -0.25)
		tmp = a / (k * k);
	elseif (m <= 7.5e+15)
		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, -0.25], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.5e+15], 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 < -0.25

   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 38.4%

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

   5. Taylor expanded in k around inf 64.7%

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

      1. unpow2 64.7%

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

   7. Simplified 64.7%

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

   if -0.25 < m < 7.5e15

   1. Initial program 96.0%

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

      1. associate-*r/ 95.9%

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

      2. associate-+l+ 95.9%

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

      3. +-commutative 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. fma-def 95.9%

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

      6. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in m around 0 91.6%

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

   5. Taylor expanded in k around inf 89.5%

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

      1. unpow2 89.5%

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

   7. Simplified 89.5%

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

   if 7.5e15 < m

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. associate-+l+ 73.1%

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

      3. +-commutative 73.1%

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

      4. distribute-rgt-out 73.1%

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

      5. fma-def 73.1%

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

      6. +-commutative 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in m around 0 2.5%

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

   5. Taylor expanded in k around 0 9.8%

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

   6. Taylor expanded in k around inf 17.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10: 57.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;a \cdot \frac{--1}{k \cdot k}\\ \mathbf{elif}\;m \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.31)
   (* a (/ (- -1.0) (* k k)))
   (if (<= m 8.2e+15) (/ a (+ 1.0 (* k k))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.31) {
		tmp = a * (-(-1.0) / (k * k));
	} else if (m <= 8.2e+15) {
		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 <= (-0.31d0)) then
        tmp = a * (-(-1.0d0) / (k * k))
    else if (m <= 8.2d+15) 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 <= -0.31) {
		tmp = a * (-(-1.0) / (k * k));
	} else if (m <= 8.2e+15) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.31:
		tmp = a * (-(-1.0) / (k * k))
	elif m <= 8.2e+15:
		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 <= -0.31)
		tmp = Float64(a * Float64(Float64(-(-1.0)) / Float64(k * k)));
	elseif (m <= 8.2e+15)
		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 <= -0.31)
		tmp = a * (-(-1.0) / (k * k));
	elseif (m <= 8.2e+15)
		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, -0.31], N[(a * N[((--1.0) / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8.2e+15], 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 < -0.309999999999999998

   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 38.4%

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

   5. Taylor expanded in k around inf 64.7%

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

      1. unpow2 64.7%

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

   7. Simplified 64.7%

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

      1. frac-2neg 64.7%

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

      2. div-inv 65.9%

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

      3. distribute-rgt-neg-in 65.9%

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

   9. Applied egg-rr 65.9%

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

   10. Taylor expanded in k around 0 65.9%

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

       1. unpow2 65.9%

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

   12. Simplified 65.9%

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

   if -0.309999999999999998 < m < 8.2e15

   1. Initial program 96.0%

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

      1. associate-*r/ 95.9%

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

      2. associate-+l+ 95.9%

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

      3. +-commutative 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. fma-def 95.9%

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

      6. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in m around 0 91.6%

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

   5. Taylor expanded in k around inf 89.5%

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

      1. unpow2 89.5%

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

   7. Simplified 89.5%

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

   if 8.2e15 < m

   1. Initial program 73.1%

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

      1. associate-*r/ 73.1%

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

      2. associate-+l+ 73.1%

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

      3. +-commutative 73.1%

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

      4. distribute-rgt-out 73.1%

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

      5. fma-def 73.1%

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

      6. +-commutative 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in m around 0 2.5%

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

   5. Taylor expanded in k around 0 9.8%

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

   6. Taylor expanded in k around inf 17.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;a \cdot \frac{--1}{k \cdot k}\\ \mathbf{elif}\;m \leq 8.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11: 46.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-288} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 5.2e-288) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 5.2e-288) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 5.2d-288) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 5.2e-288) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 5.2e-288) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 5.2e-288) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 5.2e-288) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 5.2e-288], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 5.19999999999999979e-288 or 1 < k

   1. Initial program 86.8%

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

      1. associate-*r/ 86.7%

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

      2. associate-+l+ 86.7%

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

      3. +-commutative 86.7%

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

      4. distribute-rgt-out 86.7%

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

      5. fma-def 86.7%

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

      6. +-commutative 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in m around 0 43.8%

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

   5. Taylor expanded in k around inf 49.4%

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

      1. unpow2 49.4%

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

   7. Simplified 49.4%

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

   if 5.19999999999999979e-288 < 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 55.9%

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

   5. Taylor expanded in k around 0 54.0%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-288} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12: 47.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-289}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 9.5e-289) (/ a (* k k)) (if (<= k 1.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 9.5e-289) {
		tmp = a / (k * k);
	} 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 <= 9.5d-289) then
        tmp = a / (k * k)
    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 <= 9.5e-289) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 9.5e-289:
		tmp = a / (k * k)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 9.5e-289)
		tmp = Float64(a / Float64(k * k));
	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 <= 9.5e-289)
		tmp = a / (k * k);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 9.5e-289], N[(a / N[(k * k), $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 < 9.4999999999999995e-289

   1. Initial program 88.8%

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

      1. associate-*r/ 88.8%

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

      2. associate-+l+ 88.8%

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

      3. +-commutative 88.8%

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

      4. distribute-rgt-out 88.8%

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

      5. fma-def 88.8%

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

      6. +-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in m around 0 20.2%

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

   5. Taylor expanded in k around inf 34.3%

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

      1. unpow2 34.3%

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

   7. Simplified 34.3%

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

   if 9.4999999999999995e-289 < 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 55.9%

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

   5. Taylor expanded in k around 0 54.0%

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

   if 1 < k

   1. Initial program 85.3%

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

   2. Taylor expanded in k around inf 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

      1. *-commutative 84.4%

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

      2. add-cube-cbrt 84.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{k \cdot k} \]

      3. times-frac 98.7%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k}} \]

      4. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      5. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{k} \]

      6. *-commutative 98.7%

         \[\leadsto \frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{\color{blue}{a \cdot {k}^{m}}}}{k} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}} \]
   7. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{k}}{k}} \]

      2. associate-*r/ 98.7%

         \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}}{k} \]

      3. unpow2 98.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right)} \cdot \sqrt[3]{a \cdot {k}^{m}}}{k}}{k} \]

      4. add-cube-cbrt 99.0%

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

   8. Applied egg-rr 99.0%

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

   9. Taylor expanded in m around 0 60.7%

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

       1. unpow2 60.7%

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

       2. associate-/r* 61.1%

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

   11. Simplified 61.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-289}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13: 23.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k -2e-310) (* -10.0 (* k a)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2e-310) {
		tmp = -10.0 * (k * a);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-2d-310)) then
        tmp = (-10.0d0) * (k * a)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2e-310) {
		tmp = -10.0 * (k * a);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2e-310:
		tmp = -10.0 * (k * a)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2e-310)
		tmp = Float64(-10.0 * Float64(k * a));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2e-310)
		tmp = -10.0 * (k * a);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2e-310], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -1.999999999999994e-310

   1. Initial program 88.5%

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

      1. associate-*r/ 88.5%

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

      2. associate-+l+ 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-out 88.5%

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

      5. fma-def 88.5%

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

      6. +-commutative 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in m around 0 20.7%

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

   5. Taylor expanded in k around 0 10.8%

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

   6. Taylor expanded in k around inf 18.1%

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

   if -1.999999999999994e-310 < k

   1. Initial program 91.1%

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

      1. associate-*r/ 91.1%

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

      2. associate-+l+ 91.1%

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

      3. +-commutative 91.1%

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

      4. distribute-rgt-out 91.1%

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

      5. fma-def 91.1%

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

      6. +-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in m around 0 58.6%

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

   5. Taylor expanded in k around 0 23.6%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

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 90.3%

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

   1. associate-*r/ 90.3%

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

   2. associate-+l+ 90.3%

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

   3. +-commutative 90.3%

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

   4. distribute-rgt-out 90.3%

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

   5. fma-def 90.3%

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

   6. +-commutative 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in m around 0 47.0%

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

5. Taylor expanded in k around 0 17.5%

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

6. Final simplification 17.5%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023240 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))