Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 99.1%

Time: 9.7s

Alternatives: 14

Speedup: 1.1×

• 23.1% 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: 89.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(N[Power[k, m], $MachinePrecision] * a), $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 < 1

   1. Initial program 95.6%

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

      1. associate-*r/ 95.6%

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

      2. *-commutative 95.6%

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

      3. sqr-neg 95.6%

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

      4. associate-+l+ 95.6%

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

      5. +-commutative 95.6%

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

      6. sqr-neg 95.6%

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

      7. distribute-rgt-out 95.6%

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

      8. fma-def 95.6%

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

      9. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 98.8%

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

   if 1 < k

   1. Initial program 83.4%

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

      1. associate-*r/ 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-+l+ 83.4%

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

      5. +-commutative 83.4%

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

      6. sqr-neg 83.4%

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

      7. distribute-rgt-out 83.4%

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

      8. fma-def 83.4%

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

      9. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in k around inf 82.7%

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

      1. unpow2 82.7%

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

   6. Simplified 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-/r* 95.3%

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

      3. associate-*r/ 99.2%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 99.0%

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

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-9} \lor \neg \left(m \leq 1.75 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2.8e-9) (not (<= m 1.75e-7)))
   (* (pow k m) a)
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.8e-9) || !(m <= 1.75e-7)) {
		tmp = pow(k, m) * a;
	} 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 <= (-2.8d-9)) .or. (.not. (m <= 1.75d-7))) then
        tmp = (k ** m) * a
    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 <= -2.8e-9) || !(m <= 1.75e-7)) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -2.8e-9) or not (m <= 1.75e-7):
		tmp = math.pow(k, m) * a
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2.8e-9) || !(m <= 1.75e-7))
		tmp = Float64((k ^ m) * a);
	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 <= -2.8e-9) || ~((m <= 1.75e-7)))
		tmp = (k ^ m) * a;
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2.8e-9], N[Not[LessEqual[m, 1.75e-7]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $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 < -2.79999999999999984e-9 or 1.74999999999999992e-7 < m

   1. Initial program 89.2%

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

      1. associate-*r/ 89.2%

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

      2. *-commutative 89.2%

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

      3. sqr-neg 89.2%

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

      4. associate-+l+ 89.2%

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

      5. +-commutative 89.2%

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

      6. sqr-neg 89.2%

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

      7. distribute-rgt-out 89.2%

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

      8. fma-def 89.2%

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

      9. +-commutative 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in k around 0 99.6%

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

   if -2.79999999999999984e-9 < m < 1.74999999999999992e-7

   1. Initial program 95.2%

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

      1. associate-*r/ 95.2%

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

      2. *-commutative 95.2%

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

      3. sqr-neg 95.2%

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

      4. associate-+l+ 95.2%

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

      5. +-commutative 95.2%

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

      6. sqr-neg 95.2%

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

      7. distribute-rgt-out 95.2%

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

      8. fma-def 95.2%

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

      9. +-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in m around 0 94.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-9} \lor \neg \left(m \leq 1.75 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = math.pow(k, m) * a
	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((k ^ m) * a);
	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 = (k ^ m) * a;
	else
		tmp = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(N[Power[k, m], $MachinePrecision] * a), $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 95.6%

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

      1. associate-*r/ 95.6%

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

      2. *-commutative 95.6%

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

      3. sqr-neg 95.6%

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

      4. associate-+l+ 95.6%

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

      5. +-commutative 95.6%

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

      6. sqr-neg 95.6%

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

      7. distribute-rgt-out 95.6%

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

      8. fma-def 95.6%

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

      9. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 98.8%

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

   if 1 < k

   1. Initial program 83.4%

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

      1. associate-*r/ 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-+l+ 83.4%

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

      5. +-commutative 83.4%

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

      6. sqr-neg 83.4%

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

      7. distribute-rgt-out 83.4%

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

      8. fma-def 83.4%

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

      9. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in k around inf 82.7%

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

      1. unpow2 82.7%

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

   6. Simplified 82.7%

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

      1. expm1-log1p-u 70.1%

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

      2. expm1-udef 58.5%

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

      3. *-commutative 58.5%

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

      4. clear-num 58.5%

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

      5. un-div-inv 58.5%

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

      6. pow2 58.5%

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

      7. pow-div 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. expm1-def 77.3%

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

      2. expm1-log1p 95.2%

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

   10. Simplified 95.2%

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

4. Final simplification 97.5%

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

Alternative 4: 48.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;m \leq -0.0245:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{elif}\;m \leq 2900000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (+ 1.0 (* k 10.0)))))
   (if (<= m -0.0245)
     (/ a (* k k))
     (if (<= m 1.8e-191)
       t_0
       (if (<= m 2.5e-73)
         (/ 1.0 (/ k (/ a k)))
         (if (<= m 2900000000000.0) t_0 (* k (* a -10.0))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -0.0245) {
		tmp = a / (k * k);
	} else if (m <= 1.8e-191) {
		tmp = t_0;
	} else if (m <= 2.5e-73) {
		tmp = 1.0 / (k / (a / k));
	} else if (m <= 2900000000000.0) {
		tmp = t_0;
	} else {
		tmp = k * (a * -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) :: t_0
    real(8) :: tmp
    t_0 = a / (1.0d0 + (k * 10.0d0))
    if (m <= (-0.0245d0)) then
        tmp = a / (k * k)
    else if (m <= 1.8d-191) then
        tmp = t_0
    else if (m <= 2.5d-73) then
        tmp = 1.0d0 / (k / (a / k))
    else if (m <= 2900000000000.0d0) then
        tmp = t_0
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -0.0245) {
		tmp = a / (k * k);
	} else if (m <= 1.8e-191) {
		tmp = t_0;
	} else if (m <= 2.5e-73) {
		tmp = 1.0 / (k / (a / k));
	} else if (m <= 2900000000000.0) {
		tmp = t_0;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (1.0 + (k * 10.0))
	tmp = 0
	if m <= -0.0245:
		tmp = a / (k * k)
	elif m <= 1.8e-191:
		tmp = t_0
	elif m <= 2.5e-73:
		tmp = 1.0 / (k / (a / k))
	elif m <= 2900000000000.0:
		tmp = t_0
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(1.0 + Float64(k * 10.0)))
	tmp = 0.0
	if (m <= -0.0245)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.8e-191)
		tmp = t_0;
	elseif (m <= 2.5e-73)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	elseif (m <= 2900000000000.0)
		tmp = t_0;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (1.0 + (k * 10.0));
	tmp = 0.0;
	if (m <= -0.0245)
		tmp = a / (k * k);
	elseif (m <= 1.8e-191)
		tmp = t_0;
	elseif (m <= 2.5e-73)
		tmp = 1.0 / (k / (a / k));
	elseif (m <= 2900000000000.0)
		tmp = t_0;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.0245], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.8e-191], t$95$0, If[LessEqual[m, 2.5e-73], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2900000000000.0], t$95$0, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.024500000000000001

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 28.4%

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

   5. Taylor expanded in k around inf 60.2%

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

      1. unpow2 60.2%

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

   7. Simplified 60.2%

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

   if -0.024500000000000001 < m < 1.8000000000000001e-191 or 2.4999999999999999e-73 < m < 2.9e12

   1. Initial program 95.6%

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

      1. associate-*r/ 95.6%

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

      2. *-commutative 95.6%

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

      3. sqr-neg 95.6%

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

      4. associate-+l+ 95.6%

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

      5. +-commutative 95.6%

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

      6. sqr-neg 95.6%

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

      7. distribute-rgt-out 95.6%

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

      8. fma-def 95.6%

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

      9. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in m around 0 90.7%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

   if 1.8000000000000001e-191 < m < 2.4999999999999999e-73

   1. Initial program 88.4%

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

      1. associate-*r/ 88.2%

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

      2. *-commutative 88.2%

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

      3. sqr-neg 88.2%

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

      4. associate-+l+ 88.2%

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

      5. +-commutative 88.2%

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

      6. sqr-neg 88.2%

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

      7. distribute-rgt-out 88.3%

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

      8. fma-def 88.3%

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

      9. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in m around 0 88.4%

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

   5. Taylor expanded in k around inf 66.7%

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

      1. unpow2 66.7%

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

   7. Simplified 66.7%

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

      1. clear-num 66.6%

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

      2. inv-pow 66.6%

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

   9. Applied egg-rr 66.6%

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

       1. unpow-1 66.6%

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

       2. associate-/l* 78.2%

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

   11. Simplified 78.2%

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

   if 2.9e12 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0245:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 2.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{elif}\;m \leq 2900000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 5: 44.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.5 \cdot 10^{-240}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-244}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.5e-240)
   (* a (/ 1.0 (* k k)))
   (if (<= m 9.5e-244)
     a
     (if (<= m 240000000000.0) (/ 1.0 (/ k (/ a k))) (* k (* a -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.5e-240) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 9.5e-244) {
		tmp = a;
	} else if (m <= 240000000000.0) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = k * (a * -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 <= (-5.5d-240)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 9.5d-244) then
        tmp = a
    else if (m <= 240000000000.0d0) then
        tmp = 1.0d0 / (k / (a / k))
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.5e-240) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 9.5e-244) {
		tmp = a;
	} else if (m <= 240000000000.0) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.5e-240:
		tmp = a * (1.0 / (k * k))
	elif m <= 9.5e-244:
		tmp = a
	elif m <= 240000000000.0:
		tmp = 1.0 / (k / (a / k))
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.5e-240)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 9.5e-244)
		tmp = a;
	elseif (m <= 240000000000.0)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.5e-240)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 9.5e-244)
		tmp = a;
	elseif (m <= 240000000000.0)
		tmp = 1.0 / (k / (a / k));
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.5e-240], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9.5e-244], a, If[LessEqual[m, 240000000000.0], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.49999999999999957e-240

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in m around 0 58.9%

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

      1. unpow2 58.9%

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

   9. Simplified 58.9%

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

   if -5.49999999999999957e-240 < m < 9.4999999999999995e-244

   1. Initial program 96.1%

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

      1. associate-*r/ 96.0%

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

      2. *-commutative 96.0%

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

      3. sqr-neg 96.0%

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

      4. associate-+l+ 96.0%

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

      5. +-commutative 96.0%

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

      6. sqr-neg 96.0%

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

      7. distribute-rgt-out 96.0%

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

      8. fma-def 96.0%

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

      9. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in m around 0 96.1%

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

   5. Taylor expanded in k around 0 60.9%

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

   if 9.4999999999999995e-244 < m < 2.4e11

   1. Initial program 88.7%

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

      1. associate-*r/ 88.6%

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

      2. *-commutative 88.6%

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

      3. sqr-neg 88.6%

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

      4. associate-+l+ 88.6%

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

      5. +-commutative 88.6%

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

      6. sqr-neg 88.6%

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

      7. distribute-rgt-out 88.7%

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

      8. fma-def 88.7%

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

      9. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in m around 0 81.6%

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

   5. Taylor expanded in k around inf 49.6%

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

      1. unpow2 49.6%

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

   7. Simplified 49.6%

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

      1. clear-num 49.6%

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

      2. inv-pow 49.6%

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

   9. Applied egg-rr 49.6%

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

       1. unpow-1 49.6%

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

       2. associate-/l* 57.8%

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

   11. Simplified 57.8%

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

   if 2.4e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.5 \cdot 10^{-240}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-244}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 6: 59.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8)
   (/ a (* k k))
   (if (<= m 240000000000.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8) {
		tmp = a / (k * k);
	} else if (m <= 240000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * -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 <= (-6.8d0)) then
        tmp = a / (k * k)
    else if (m <= 240000000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8) {
		tmp = a / (k * k);
	} else if (m <= 240000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.8:
		tmp = a / (k * k)
	elif m <= 240000000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 240000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.8)
		tmp = a / (k * k);
	elseif (m <= 240000000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 240000000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.79999999999999982

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 28.7%

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

   5. Taylor expanded in k around inf 60.9%

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

      1. unpow2 60.9%

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

   7. Simplified 60.9%

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

   if -6.79999999999999982 < m < 2.4e11

   1. Initial program 94.3%

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

      1. associate-*r/ 94.2%

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

      2. *-commutative 94.2%

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

      3. sqr-neg 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. sqr-neg 94.2%

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

      7. distribute-rgt-out 94.2%

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

      8. fma-def 94.2%

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

      9. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 89.3%

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

   if 2.4e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 7: 43.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.4 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-191}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 330000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.4e-238)
     t_0
     (if (<= m 1.75e-191)
       a
       (if (<= m 330000000000.0) t_0 (* k (* a -10.0)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.4e-238) {
		tmp = t_0;
	} else if (m <= 1.75e-191) {
		tmp = a;
	} else if (m <= 330000000000.0) {
		tmp = t_0;
	} else {
		tmp = k * (a * -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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-1.4d-238)) then
        tmp = t_0
    else if (m <= 1.75d-191) then
        tmp = a
    else if (m <= 330000000000.0d0) then
        tmp = t_0
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.4e-238) {
		tmp = t_0;
	} else if (m <= 1.75e-191) {
		tmp = a;
	} else if (m <= 330000000000.0) {
		tmp = t_0;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -1.4e-238:
		tmp = t_0
	elif m <= 1.75e-191:
		tmp = a
	elif m <= 330000000000.0:
		tmp = t_0
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.4e-238)
		tmp = t_0;
	elseif (m <= 1.75e-191)
		tmp = a;
	elseif (m <= 330000000000.0)
		tmp = t_0;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -1.4e-238)
		tmp = t_0;
	elseif (m <= 1.75e-191)
		tmp = a;
	elseif (m <= 330000000000.0)
		tmp = t_0;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.4e-238], t$95$0, If[LessEqual[m, 1.75e-191], a, If[LessEqual[m, 330000000000.0], t$95$0, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.40000000000000002e-238 or 1.75000000000000003e-191 < m < 3.3e11

   1. Initial program 97.8%

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

      1. associate-*r/ 97.7%

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

      2. *-commutative 97.7%

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

      3. sqr-neg 97.7%

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

      4. associate-+l+ 97.7%

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

      5. +-commutative 97.7%

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

      6. sqr-neg 97.7%

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

      7. distribute-rgt-out 97.8%

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

      8. fma-def 97.8%

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

      9. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in m around 0 51.8%

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

   5. Taylor expanded in k around inf 57.5%

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

      1. unpow2 57.5%

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

   7. Simplified 57.5%

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

   if -1.40000000000000002e-238 < m < 1.75000000000000003e-191

   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. *-commutative 94.0%

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

      3. sqr-neg 94.0%

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

      4. associate-+l+ 94.0%

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

      5. +-commutative 94.0%

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

      6. sqr-neg 94.0%

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

      7. distribute-rgt-out 94.0%

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

      8. fma-def 94.0%

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

      9. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in m around 0 94.1%

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

   5. Taylor expanded in k around 0 58.7%

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

   if 3.3e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-238}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-191}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 330000000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 8: 44.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.3e-230)
   (/ a (* k k))
   (if (<= m 1.6e-248)
     a
     (if (<= m 240000000000.0) (/ (/ a k) k) (* k (* a -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.3e-230) {
		tmp = a / (k * k);
	} else if (m <= 1.6e-248) {
		tmp = a;
	} else if (m <= 240000000000.0) {
		tmp = (a / k) / k;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.3d-230)) then
        tmp = a / (k * k)
    else if (m <= 1.6d-248) then
        tmp = a
    else if (m <= 240000000000.0d0) then
        tmp = (a / k) / k
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.3e-230) {
		tmp = a / (k * k);
	} else if (m <= 1.6e-248) {
		tmp = a;
	} else if (m <= 240000000000.0) {
		tmp = (a / k) / k;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.3e-230:
		tmp = a / (k * k)
	elif m <= 1.6e-248:
		tmp = a
	elif m <= 240000000000.0:
		tmp = (a / k) / k
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.3e-230)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.6e-248)
		tmp = a;
	elseif (m <= 240000000000.0)
		tmp = Float64(Float64(a / k) / k);
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.3e-230)
		tmp = a / (k * k);
	elseif (m <= 1.6e-248)
		tmp = a;
	elseif (m <= 240000000000.0)
		tmp = (a / k) / k;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.3e-230], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.6e-248], a, If[LessEqual[m, 240000000000.0], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -1.3000000000000001e-230

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 44.8%

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

   5. Taylor expanded in k around inf 58.9%

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

      1. unpow2 58.9%

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

   7. Simplified 58.9%

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

   if -1.3000000000000001e-230 < m < 1.60000000000000009e-248

   1. Initial program 96.1%

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

      1. associate-*r/ 96.0%

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

      2. *-commutative 96.0%

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

      3. sqr-neg 96.0%

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

      4. associate-+l+ 96.0%

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

      5. +-commutative 96.0%

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

      6. sqr-neg 96.0%

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

      7. distribute-rgt-out 96.0%

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

      8. fma-def 96.0%

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

      9. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in m around 0 96.1%

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

   5. Taylor expanded in k around 0 60.9%

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

   if 1.60000000000000009e-248 < m < 2.4e11

   1. Initial program 88.7%

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

      1. associate-*r/ 88.6%

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

      2. *-commutative 88.6%

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

      3. sqr-neg 88.6%

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

      4. associate-+l+ 88.6%

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

      5. +-commutative 88.6%

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

      6. sqr-neg 88.6%

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

      7. distribute-rgt-out 88.7%

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

      8. fma-def 88.7%

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

      9. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in m around 0 81.6%

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

   5. Taylor expanded in k around inf 49.6%

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

      1. unpow2 49.6%

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

   7. Simplified 49.6%

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

   8. Taylor expanded in a around 0 49.6%

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

      1. unpow2 49.6%

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

      2. associate-/l/ 57.7%

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

   10. Simplified 57.7%

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

   if 2.4e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-248}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 9: 44.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.9 \cdot 10^{-229}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.4 \cdot 10^{-246}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 270000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.9e-229)
   (* a (/ 1.0 (* k k)))
   (if (<= m 5.4e-246)
     a
     (if (<= m 270000000000.0) (/ (/ a k) k) (* k (* a -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.9e-229) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 5.4e-246) {
		tmp = a;
	} else if (m <= 270000000000.0) {
		tmp = (a / k) / k;
	} else {
		tmp = k * (a * -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 <= (-4.9d-229)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 5.4d-246) then
        tmp = a
    else if (m <= 270000000000.0d0) then
        tmp = (a / k) / k
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.9e-229) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 5.4e-246) {
		tmp = a;
	} else if (m <= 270000000000.0) {
		tmp = (a / k) / k;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.9e-229:
		tmp = a * (1.0 / (k * k))
	elif m <= 5.4e-246:
		tmp = a
	elif m <= 270000000000.0:
		tmp = (a / k) / k
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.9e-229)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 5.4e-246)
		tmp = a;
	elseif (m <= 270000000000.0)
		tmp = Float64(Float64(a / k) / k);
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.9e-229)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 5.4e-246)
		tmp = a;
	elseif (m <= 270000000000.0)
		tmp = (a / k) / k;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.9e-229], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.4e-246], a, If[LessEqual[m, 270000000000.0], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -4.89999999999999974e-229

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in m around 0 58.9%

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

      1. unpow2 58.9%

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

   9. Simplified 58.9%

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

   if -4.89999999999999974e-229 < m < 5.3999999999999998e-246

   1. Initial program 96.1%

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

      1. associate-*r/ 96.0%

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

      2. *-commutative 96.0%

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

      3. sqr-neg 96.0%

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

      4. associate-+l+ 96.0%

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

      5. +-commutative 96.0%

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

      6. sqr-neg 96.0%

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

      7. distribute-rgt-out 96.0%

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

      8. fma-def 96.0%

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

      9. +-commutative 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in m around 0 96.1%

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

   5. Taylor expanded in k around 0 60.9%

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

   if 5.3999999999999998e-246 < m < 2.7e11

   1. Initial program 88.7%

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

      1. associate-*r/ 88.6%

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

      2. *-commutative 88.6%

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

      3. sqr-neg 88.6%

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

      4. associate-+l+ 88.6%

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

      5. +-commutative 88.6%

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

      6. sqr-neg 88.6%

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

      7. distribute-rgt-out 88.7%

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

      8. fma-def 88.7%

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

      9. +-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in m around 0 81.6%

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

   5. Taylor expanded in k around inf 49.6%

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

      1. unpow2 49.6%

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

   7. Simplified 49.6%

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

   8. Taylor expanded in a around 0 49.6%

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

      1. unpow2 49.6%

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

      2. associate-/l/ 57.7%

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

   10. Simplified 57.7%

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

   if 2.7e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.9 \cdot 10^{-229}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.4 \cdot 10^{-246}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 270000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 10: 58.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3100000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8)
   (/ a (* k k))
   (if (<= m 3100000000000.0) (/ a (+ 1.0 (* k k))) (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8) {
		tmp = a / (k * k);
	} else if (m <= 3100000000000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = k * (a * -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 <= (-6.8d0)) then
        tmp = a / (k * k)
    else if (m <= 3100000000000.0d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8) {
		tmp = a / (k * k);
	} else if (m <= 3100000000000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.8:
		tmp = a / (k * k)
	elif m <= 3100000000000.0:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 3100000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.8)
		tmp = a / (k * k);
	elseif (m <= 3100000000000.0)
		tmp = a / (1.0 + (k * k));
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3100000000000.0], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.79999999999999982

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 28.7%

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

   5. Taylor expanded in k around inf 60.9%

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

      1. unpow2 60.9%

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

   7. Simplified 60.9%

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

   if -6.79999999999999982 < m < 3.1e12

   1. Initial program 94.3%

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

      1. associate-*r/ 94.2%

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

      2. *-commutative 94.2%

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

      3. sqr-neg 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. sqr-neg 94.2%

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

      7. distribute-rgt-out 94.2%

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

      8. fma-def 94.2%

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

      9. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 89.3%

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

   5. Taylor expanded in k around inf 87.2%

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

      1. unpow2 87.2%

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

   7. Simplified 87.2%

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

   if 3.1e12 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.8:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3100000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 11: 31.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.9e-36)
   (/ a (* k 10.0))
   (if (<= m 240000000000.0) a (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.9e-36) {
		tmp = a / (k * 10.0);
	} else if (m <= 240000000000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -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 <= (-5.9d-36)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 240000000000.0d0) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.9e-36) {
		tmp = a / (k * 10.0);
	} else if (m <= 240000000000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.9e-36:
		tmp = a / (k * 10.0)
	elif m <= 240000000000.0:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.9e-36)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 240000000000.0)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.9e-36)
		tmp = a / (k * 10.0);
	elseif (m <= 240000000000.0)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.9e-36], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 240000000000.0], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.89999999999999995e-36

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 30.0%

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

   5. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   7. Simplified 13.2%

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

   8. Taylor expanded in k around inf 22.4%

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

      1. *-commutative 22.4%

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

   10. Simplified 22.4%

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

   if -5.89999999999999995e-36 < m < 2.4e11

   1. Initial program 94.0%

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

      1. associate-*r/ 94.0%

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

      2. *-commutative 94.0%

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

      3. sqr-neg 94.0%

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

      4. associate-+l+ 94.0%

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

      5. +-commutative 94.0%

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

      6. sqr-neg 94.0%

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

      7. distribute-rgt-out 94.0%

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

      8. fma-def 94.0%

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

      9. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in m around 0 91.0%

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

   5. Taylor expanded in k around 0 47.3%

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

   if 2.4e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 240000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 12: 25.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 255000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 255000000000.0) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 255000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 255000000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 255000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 255000000000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 255000000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 255000000000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 255000000000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.55e11

   1. Initial program 97.0%

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

      1. associate-*r/ 97.0%

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

      2. *-commutative 97.0%

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

      3. sqr-neg 97.0%

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

      4. associate-+l+ 97.0%

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

      5. +-commutative 97.0%

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

      6. sqr-neg 97.0%

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

      7. distribute-rgt-out 97.0%

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

      8. fma-def 97.0%

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

      9. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in m around 0 60.1%

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

   5. Taylor expanded in k around 0 25.4%

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

   if 2.55e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 255000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13: 25.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 240000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 240000000000.0) a (* k (* a -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 240000000000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -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 <= 240000000000.0d0) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 240000000000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 240000000000.0:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 240000000000.0)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 240000000000.0)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 240000000000.0], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.4e11

   1. Initial program 97.0%

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

      1. associate-*r/ 97.0%

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

      2. *-commutative 97.0%

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

      3. sqr-neg 97.0%

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

      4. associate-+l+ 97.0%

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

      5. +-commutative 97.0%

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

      6. sqr-neg 97.0%

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

      7. distribute-rgt-out 97.0%

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

      8. fma-def 97.0%

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

      9. +-commutative 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in m around 0 60.1%

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

   5. Taylor expanded in k around 0 25.4%

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

   if 2.4e11 < m

   1. Initial program 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. sqr-neg 80.9%

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

      4. associate-+l+ 80.9%

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

      5. +-commutative 80.9%

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

      6. sqr-neg 80.9%

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

      7. distribute-rgt-out 80.9%

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

      8. fma-def 80.9%

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

      9. +-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 5.1%

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

   6. Taylor expanded in k around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. *-commutative 16.3%

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

      3. associate-*l* 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 240000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 14: 20.2% 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 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. *-commutative 91.1%

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

   3. sqr-neg 91.1%

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

   4. associate-+l+ 91.1%

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

   5. +-commutative 91.1%

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

   6. sqr-neg 91.1%

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

   7. distribute-rgt-out 91.1%

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

   8. fma-def 91.1%

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

   9. +-commutative 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in m around 0 39.2%

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

5. Taylor expanded in k around 0 17.6%

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

6. Final simplification 17.6%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023292 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))