Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 99.0%

Time: 9.5s

Alternatives: 13

Speedup: 1.1×

• 23.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.45e-5) (* a (pow k m)) (/ (* a (/ (pow k m) k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * pow(k, m);
	} else {
		tmp = (a * (pow(k, m) / 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 <= 2.45d-5) then
        tmp = a * (k ** m)
    else
        tmp = (a * ((k ** m) / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (a * (Math.pow(k, m) / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.45e-5:
		tmp = a * math.pow(k, m)
	else:
		tmp = (a * (math.pow(k, m) / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.45e-5)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.45e-5)
		tmp = a * (k ^ m);
	else
		tmp = (a * ((k ^ m) / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.45e-5], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e-5

   1. Initial program 96.5%

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

      1. associate-*r/ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. distribute-rgt-out 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in k around 0 58.4%

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

      1. exp-to-pow 99.6%

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

   6. Simplified 99.6%

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

   if 2.45e-5 < k

   1. Initial program 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-+l+ 82.4%

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

      3. +-commutative 82.4%

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

      4. distribute-rgt-out 82.4%

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

      5. fma-def 82.4%

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

      6. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in k around inf 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

      1. associate-/r* 93.8%

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

      2. div-inv 93.8%

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

   8. Applied egg-rr 93.8%

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

      1. associate-*r/ 93.8%

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

      2. *-rgt-identity 93.8%

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

   10. Simplified 93.8%

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

       1. associate-*r/ 98.8%

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

   12. Applied egg-rr 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 2: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.45e-5) (* a (pow k m)) (* a (/ (/ (pow k m) k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * ((pow(k, m) / 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 <= 2.45d-5) then
        tmp = a * (k ** m)
    else
        tmp = a * (((k ** m) / k) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * ((Math.pow(k, m) / k) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.45e-5:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * ((math.pow(k, m) / k) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.45e-5)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(Float64((k ^ m) / k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.45e-5)
		tmp = a * (k ^ m);
	else
		tmp = a * (((k ^ m) / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.45e-5], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e-5

   1. Initial program 96.5%

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

      1. associate-*r/ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. distribute-rgt-out 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in k around 0 58.4%

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

      1. exp-to-pow 99.6%

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

   6. Simplified 99.6%

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

   if 2.45e-5 < k

   1. Initial program 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-+l+ 82.4%

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

      3. +-commutative 82.4%

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

      4. distribute-rgt-out 82.4%

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

      5. fma-def 82.4%

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

      6. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in k around inf 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

      1. associate-/r* 93.8%

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

      2. div-inv 93.8%

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

   8. Applied egg-rr 93.8%

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

      1. associate-*r/ 93.8%

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

      2. *-rgt-identity 93.8%

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

   10. Simplified 93.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{{k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.9 \cdot 10^{-10} \lor \neg \left(m \leq 0.45\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -7.9e-10) (not (<= m 0.45)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -7.9e-10) || !(m <= 0.45)) {
		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 <= (-7.9d-10)) .or. (.not. (m <= 0.45d0))) 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 <= -7.9e-10) || !(m <= 0.45)) {
		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 <= -7.9e-10) or not (m <= 0.45):
		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 <= -7.9e-10) || !(m <= 0.45))
		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 <= -7.9e-10) || ~((m <= 0.45)))
		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, -7.9e-10], N[Not[LessEqual[m, 0.45]], $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 < -7.8999999999999996e-10 or 0.450000000000000011 < m

   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 k around 0 64.8%

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

      1. exp-to-pow 98.9%

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

   6. Simplified 98.9%

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

   if -7.8999999999999996e-10 < m < 0.450000000000000011

   1. Initial program 94.1%

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

      1. associate-*r/ 94.0%

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

      2. associate-+l+ 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt-out 94.0%

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

      5. fma-def 94.0%

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

      6. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in m around 0 91.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.9 \cdot 10^{-10} \lor \neg \left(m \leq 0.45\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.45e-5) (* a (pow k m)) (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	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 (k <= 2.45d-5) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.45e-5) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.45e-5:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.45e-5)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.45e-5)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.45e-5], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e-5

   1. Initial program 96.5%

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

      1. associate-*r/ 96.5%

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

      2. associate-+l+ 96.5%

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

      3. +-commutative 96.5%

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

      4. distribute-rgt-out 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in k around 0 58.4%

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

      1. exp-to-pow 99.6%

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

   6. Simplified 99.6%

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

   if 2.45e-5 < k

   1. Initial program 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-+l+ 82.4%

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

      3. +-commutative 82.4%

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

      4. distribute-rgt-out 82.4%

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

      5. fma-def 82.4%

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

      6. +-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in k around inf 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

      1. expm1-log1p-u 81.3%

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

      2. expm1-udef 63.5%

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

      3. pow2 63.5%

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

      4. pow-div 75.8%

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

   8. Applied egg-rr 75.8%

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

      1. expm1-def 93.5%

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

      2. expm1-log1p 93.5%

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

   10. Simplified 93.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 5: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0096:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.0096)
   (/ a (* k k))
   (if (<= m 3.15e+16) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0096) {
		tmp = a / (k * k);
	} else if (m <= 3.15e+16) {
		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 <= (-0.0096d0)) then
        tmp = a / (k * k)
    else if (m <= 3.15d+16) 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 <= -0.0096) {
		tmp = a / (k * k);
	} else if (m <= 3.15e+16) {
		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 <= -0.0096:
		tmp = a / (k * k)
	elif m <= 3.15e+16:
		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 <= -0.0096)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 3.15e+16)
		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 <= -0.0096)
		tmp = a / (k * k);
	elseif (m <= 3.15e+16)
		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, -0.0096], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.15e+16], 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 < -0.00959999999999999916

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. fma-def 99.0%

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

      6. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in m around 0 42.3%

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

   5. Taylor expanded in k around inf 64.9%

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

      1. unpow2 64.9%

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

   7. Simplified 64.9%

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

   if -0.00959999999999999916 < m < 3.15e16

   1. Initial program 91.0%

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

      1. associate-*r/ 91.0%

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

      2. associate-+l+ 91.0%

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

      3. +-commutative 91.0%

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

      4. distribute-rgt-out 91.0%

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

      5. fma-def 91.0%

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

      6. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 87.8%

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

   if 3.15e16 < m

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. associate-+l+ 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-rgt-out 79.5%

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

      5. fma-def 79.5%

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

      6. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 6.5%

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

   6. Taylor expanded in k around inf 19.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0096:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 6: 47.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-308} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 8e-308) (not (<= k 0.1)))
   (/ a (* k k))
   (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 8e-308) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-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 ((k <= 8d-308) .or. (.not. (k <= 0.1d0))) then
        tmp = a / (k * k)
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 8e-308) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 8e-308) or not (k <= 0.1):
		tmp = a / (k * k)
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 8e-308) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 8e-308) || ~((k <= 0.1)))
		tmp = a / (k * k);
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 8e-308], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000026e-308 or 0.10000000000000001 < k

   1. Initial program 85.5%

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

      1. associate-*r/ 85.4%

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

      2. associate-+l+ 85.4%

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

      3. +-commutative 85.4%

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

      4. distribute-rgt-out 85.4%

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

      5. fma-def 85.4%

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

      6. +-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in m around 0 49.2%

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

   5. Taylor expanded in k around inf 53.3%

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

      1. unpow2 53.3%

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

   7. Simplified 53.3%

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

   if 8.00000000000000026e-308 < 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 m around 0 39.4%

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

   5. Taylor expanded in k around 0 39.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-308} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 7: 47.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7.7e-308)
   (/ a (* k k))
   (if (<= k 0.075) (+ a (* -10.0 (* k a))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.7e-308) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = a / (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 (k <= 7.7d-308) then
        tmp = a / (k * k)
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.7e-308) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7.7e-308:
		tmp = a / (k * k)
	elif k <= 0.075:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7.7e-308)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7.7e-308)
		tmp = a / (k * k);
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7.7e-308], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.70000000000000019e-308

   1. Initial program 91.7%

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

      1. associate-*r/ 91.7%

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

      2. associate-+l+ 91.7%

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

      3. +-commutative 91.7%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 26.0%

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

   5. Taylor expanded in k around inf 39.9%

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

      1. unpow2 39.9%

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

   7. Simplified 39.9%

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

   if 7.70000000000000019e-308 < k < 0.0749999999999999972

   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 39.4%

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

   5. Taylor expanded in k around 0 39.2%

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

   if 0.0749999999999999972 < k

   1. Initial program 82.1%

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

   2. Taylor expanded in m around 0 61.7%

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

   3. Taylor expanded in k around inf 61.3%

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

      1. *-commutative 61.3%

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

   5. Simplified 61.3%

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

      1. +-commutative 61.3%

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

      2. distribute-lft-in 61.3%

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

      3. *-commutative 61.3%

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

   7. Applied egg-rr 61.3%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 47.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 41:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.58e-307)
   (/ a (* k k))
   (if (<= k 41.0) (/ a (+ 1.0 (* k 10.0))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.58e-307) {
		tmp = a / (k * k);
	} else if (k <= 41.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (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 (k <= 1.58d-307) then
        tmp = a / (k * k)
    else if (k <= 41.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.58e-307) {
		tmp = a / (k * k);
	} else if (k <= 41.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.58e-307:
		tmp = a / (k * k)
	elif k <= 41.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.58e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 41.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.58e-307)
		tmp = a / (k * k);
	elseif (k <= 41.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.58e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 41.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.57999999999999992e-307

   1. Initial program 91.7%

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

      1. associate-*r/ 91.7%

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

      2. associate-+l+ 91.7%

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

      3. +-commutative 91.7%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 26.0%

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

   5. Taylor expanded in k around inf 39.9%

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

      1. unpow2 39.9%

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

   7. Simplified 39.9%

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

   if 1.57999999999999992e-307 < k < 41

   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 39.0%

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

   5. Taylor expanded in k around 0 38.9%

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

      1. *-commutative 38.9%

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

   7. Simplified 38.9%

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

   if 41 < k

   1. Initial program 81.9%

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

   2. Taylor expanded in m around 0 62.2%

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

   3. Taylor expanded in k around inf 61.8%

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

      1. *-commutative 61.8%

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

   5. Simplified 61.8%

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

      1. +-commutative 61.8%

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

      2. distribute-lft-in 61.8%

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

      3. *-commutative 61.8%

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

   7. Applied egg-rr 61.8%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 41:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9: 57.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0096:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.0096)
   (/ a (* k k))
   (if (<= m 3.15e+16) (/ a (+ 1.0 (* k k))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0096) {
		tmp = a / (k * k);
	} else if (m <= 3.15e+16) {
		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 <= (-0.0096d0)) then
        tmp = a / (k * k)
    else if (m <= 3.15d+16) 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 <= -0.0096) {
		tmp = a / (k * k);
	} else if (m <= 3.15e+16) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.0096:
		tmp = a / (k * k)
	elif m <= 3.15e+16:
		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 <= -0.0096)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 3.15e+16)
		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 <= -0.0096)
		tmp = a / (k * k);
	elseif (m <= 3.15e+16)
		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, -0.0096], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.15e+16], 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 < -0.00959999999999999916

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. fma-def 99.0%

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

      6. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in m around 0 42.3%

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

   5. Taylor expanded in k around inf 64.9%

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

      1. unpow2 64.9%

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

   7. Simplified 64.9%

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

   if -0.00959999999999999916 < m < 3.15e16

   1. Initial program 91.0%

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

      1. associate-*r/ 91.0%

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

      2. associate-+l+ 91.0%

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

      3. +-commutative 91.0%

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

      4. distribute-rgt-out 91.0%

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

      5. fma-def 91.0%

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

      6. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 87.8%

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

   5. Taylor expanded in k around inf 85.8%

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

      1. unpow2 85.8%

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

   7. Simplified 85.8%

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

   if 3.15e16 < m

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. associate-+l+ 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-rgt-out 79.5%

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

      5. fma-def 79.5%

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

      6. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 6.5%

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

   6. Taylor expanded in k around inf 19.1%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0096:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10: 46.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-307} \lor \neg \left(k \leq 41\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.8e-307) (not (<= k 41.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.8e-307) || !(k <= 41.0)) {
		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 <= 1.8d-307) .or. (.not. (k <= 41.0d0))) 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 <= 1.8e-307) || !(k <= 41.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.8e-307) or not (k <= 41.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.8e-307) || !(k <= 41.0))
		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 <= 1.8e-307) || ~((k <= 41.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.8e-307], N[Not[LessEqual[k, 41.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000003e-307 or 41 < k

   1. Initial program 85.4%

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

      1. associate-*r/ 85.4%

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

      2. associate-+l+ 85.4%

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

      3. +-commutative 85.4%

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

      4. distribute-rgt-out 85.4%

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

      5. fma-def 85.4%

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

      6. +-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in m around 0 49.5%

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

   5. Taylor expanded in k around inf 53.6%

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

      1. unpow2 53.6%

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

   7. Simplified 53.6%

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

   if 1.80000000000000003e-307 < k < 41

   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 39.0%

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

   5. Taylor expanded in k around 0 38.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-307} \lor \neg \left(k \leq 41\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 30.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.6e-9) (/ a (* k 10.0)) (if (<= m 6.6e+16) a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.6e-9) {
		tmp = a / (k * 10.0);
	} else if (m <= 6.6e+16) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.6d-9)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 6.6d+16) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.6e-9) {
		tmp = a / (k * 10.0);
	} else if (m <= 6.6e+16) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.6e-9:
		tmp = a / (k * 10.0)
	elif m <= 6.6e+16:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.6e-9)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 6.6e+16)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.6e-9)
		tmp = a / (k * 10.0);
	elseif (m <= 6.6e+16)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.6e-9], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.6e+16], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.5999999999999998e-9

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. fma-def 99.0%

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

      6. +-commutative 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in m around 0 42.3%

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

   5. Taylor expanded in k around 0 16.6%

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

      1. *-commutative 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in k around inf 20.5%

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

      1. *-commutative 20.5%

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

   10. Simplified 20.5%

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

   if -4.5999999999999998e-9 < m < 6.6e16

   1. Initial program 90.9%

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

      1. associate-*r/ 90.9%

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

      2. associate-+l+ 90.9%

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

      3. +-commutative 90.9%

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

      4. distribute-rgt-out 90.9%

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

      5. fma-def 90.9%

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

      6. +-commutative 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in m around 0 88.2%

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

   5. Taylor expanded in k around 0 38.8%

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

   if 6.6e16 < m

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. associate-+l+ 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-rgt-out 79.5%

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

      5. fma-def 79.5%

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

      6. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 6.5%

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

   6. Taylor expanded in k around inf 19.1%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 12: 24.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 3.15e+16) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.15e+16) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 3.15d+16) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.15e+16) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 3.15e+16:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.15e+16)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 3.15e+16)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.15e+16], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.15e16

   1. Initial program 95.0%

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

      1. associate-*r/ 95.0%

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

      2. associate-+l+ 95.0%

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

      3. +-commutative 95.0%

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

      4. distribute-rgt-out 95.0%

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

      5. fma-def 95.0%

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

      6. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in m around 0 64.8%

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

   5. Taylor expanded in k around 0 21.1%

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

   if 3.15e16 < m

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. associate-+l+ 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-rgt-out 79.5%

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

      5. fma-def 79.5%

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

      6. +-commutative 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 6.5%

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

   6. Taylor expanded in k around inf 19.1%

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

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.15 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13: 19.9% 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 45.9%

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

5. Taylor expanded in k around 0 15.8%

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

6. Final simplification 15.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023193 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))