Falkner and Boettcher, Appendix A

Percentage Accurate: 90.7% → 97.2%

Time: 11.8s

Alternatives: 12

Speedup: 1.0×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.1:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) / k) * (a / (k + 10.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   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-*l/ 91.8%

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

      2. sqr-neg 91.8%

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

      3. associate-+l+ 91.8%

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

      4. sqr-neg 91.8%

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

      5. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in k around 0 99.6%

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

   if 0.10000000000000001 < k

   1. Initial program 85.3%

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

      1. associate-*l/ 84.3%

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

      2. sqr-neg 84.3%

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

      3. associate-+l+ 84.3%

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

      4. sqr-neg 84.3%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

      4. +-commutative 84.2%

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

      5. fma-define 84.2%

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

      6. +-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

   7. Taylor expanded in k around inf 82.6%

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

      1. +-commutative 82.6%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 82.6%

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

      3. distribute-rgt-in 82.6%

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

   9. Simplified 82.6%

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

       1. associate-/l* 89.3%

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

       2. associate-/r/ 97.2%

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

   11. Applied egg-rr 97.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-10} \lor \neg \left(m \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2.6e-10) (not (<= m 4.3e-7)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.6e-10) || !(m <= 4.3e-7)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-2.6d-10)) .or. (.not. (m <= 4.3d-7))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.6e-10) || !(m <= 4.3e-7)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -2.6e-10) or not (m <= 4.3e-7):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2.6e-10) || !(m <= 4.3e-7))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -2.6e-10) || ~((m <= 4.3e-7)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2.6e-10], N[Not[LessEqual[m, 4.3e-7]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.59999999999999981e-10 or 4.3000000000000001e-7 < m

   1. Initial program 89.3%

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

      1. associate-*l/ 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l+ 86.3%

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

      4. sqr-neg 86.3%

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

      5. distribute-rgt-out 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in k around 0 99.4%

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

   if -2.59999999999999981e-10 < m < 4.3000000000000001e-7

   1. Initial program 93.9%

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

      1. associate-*l/ 93.9%

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

      2. sqr-neg 93.9%

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

      3. associate-+l+ 93.9%

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

      4. sqr-neg 93.9%

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

      5. distribute-rgt-out 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in m around 0 92.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.6 \cdot 10^{-10} \lor \neg \left(m \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.15 \cdot 10^{+19} \lor \neg \left(k \leq 0.076\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -1.15e+19) (not (<= k 0.076)))
   (/ a (* k (+ k 10.0)))
   (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.15e+19) || !(k <= 0.076)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-1.15d+19)) .or. (.not. (k <= 0.076d0))) then
        tmp = a / (k * (k + 10.0d0))
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.15e+19) || !(k <= 0.076)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -1.15e+19) or not (k <= 0.076):
		tmp = a / (k * (k + 10.0))
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -1.15e+19) || !(k <= 0.076))
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -1.15e+19) || ~((k <= 0.076)))
		tmp = a / (k * (k + 10.0));
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -1.15e+19], N[Not[LessEqual[k, 0.076]], $MachinePrecision]], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -1.15e19 or 0.0759999999999999981 < k

   1. Initial program 83.7%

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

      1. associate-*l/ 80.2%

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

      2. sqr-neg 80.2%

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

      3. associate-+l+ 80.2%

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

      4. sqr-neg 80.2%

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

      5. distribute-rgt-out 80.2%

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

   3. Simplified 80.2%

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

      1. *-commutative 80.2%

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

      2. clear-num 80.2%

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

      3. un-div-inv 80.2%

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

      4. +-commutative 80.2%

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

      5. fma-define 80.2%

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

      6. +-commutative 80.2%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in k around inf 79.1%

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

      1. +-commutative 79.1%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 79.1%

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

      3. distribute-rgt-in 79.1%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in m around 0 55.1%

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

   if -1.15e19 < k < 0.0759999999999999981

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.8%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.15 \cdot 10^{+19} \lor \neg \left(k \leq 0.076\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -10.0) || !(k <= 1.75)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-10.0d0)) .or. (.not. (k <= 1.75d0))) then
        tmp = a / (k * (k + 10.0d0))
    else
        tmp = a / (1.0d0 + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -10.0) || !(k <= 1.75)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -10.0) or not (k <= 1.75):
		tmp = a / (k * (k + 10.0))
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -10.0) || !(k <= 1.75))
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -10.0) || ~((k <= 1.75)))
		tmp = a / (k * (k + 10.0));
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -10.0], N[Not[LessEqual[k, 1.75]], $MachinePrecision]], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -10 or 1.75 < 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-*l/ 79.8%

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

      2. sqr-neg 79.8%

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

      3. associate-+l+ 79.8%

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

      4. sqr-neg 79.8%

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

      5. distribute-rgt-out 79.8%

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

   3. Simplified 79.8%

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

      1. *-commutative 79.8%

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

      2. clear-num 79.8%

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

      3. un-div-inv 79.8%

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

      4. +-commutative 79.8%

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

      5. fma-define 79.8%

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

      6. +-commutative 79.8%

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

   6. Applied egg-rr 79.8%

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

   7. Taylor expanded in k around inf 79.2%

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

      1. +-commutative 79.2%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 79.2%

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

      3. distribute-rgt-in 79.2%

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

   9. Simplified 79.2%

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

   10. Taylor expanded in m around 0 56.0%

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

   if -10 < k < 1.75

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.2%

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

      1. *-commutative 37.2%

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

   8. Simplified 37.2%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -10 \lor \neg \left(k \leq 1.75\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.7:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.7)
   (/ a (* k (+ k 10.0)))
   (if (<= k 1.75) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) (+ k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.7d0)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= 1.75d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / (k + 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.7:
		tmp = a / (k * (k + 10.0))
	elif k <= 1.75:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / (k + 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.7)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= 1.75)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / Float64(k + 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.7)
		tmp = a / (k * (k + 10.0));
	elseif (k <= 1.75)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / (k + 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.7], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.75], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.69999999999999996

   1. Initial program 80.4%

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

      1. associate-*l/ 71.7%

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

      2. sqr-neg 71.7%

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

      3. associate-+l+ 71.7%

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

      4. sqr-neg 71.7%

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

      5. distribute-rgt-out 71.7%

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

   3. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. clear-num 71.7%

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

      3. un-div-inv 71.7%

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

      4. +-commutative 71.7%

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

      5. fma-define 71.7%

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

      6. +-commutative 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in k around inf 71.7%

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

      1. +-commutative 71.7%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 71.7%

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

      3. distribute-rgt-in 71.7%

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

   9. Simplified 71.7%

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

   10. Taylor expanded in m around 0 44.8%

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

   if -1.69999999999999996 < k < 1.75

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.2%

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

      1. *-commutative 37.2%

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

   8. Simplified 37.2%

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

   if 1.75 < k

   1. Initial program 84.8%

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

      1. associate-*l/ 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-+l+ 83.8%

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

      4. sqr-neg 83.8%

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

      5. distribute-rgt-out 83.8%

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

   3. Simplified 83.8%

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

      1. *-commutative 83.8%

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

      2. clear-num 83.7%

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

      3. un-div-inv 83.7%

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

      4. +-commutative 83.7%

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

      5. fma-define 83.7%

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

      6. +-commutative 83.7%

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

   6. Applied egg-rr 83.7%

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

   7. Taylor expanded in k around inf 82.8%

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

      1. +-commutative 82.8%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 82.8%

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

      3. distribute-rgt-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in m around 0 61.6%

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

       1. associate-/r* 62.6%

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

       2. +-commutative 62.6%

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

   12. Simplified 62.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.35:\\ \;\;\;\;\frac{1}{\frac{k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.35)
   (/ 1.0 (/ (* k (+ k 10.0)) a))
   (if (<= k 1.75) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) (+ k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.35) {
		tmp = 1.0 / ((k * (k + 10.0)) / a);
	} else if (k <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.35d0)) then
        tmp = 1.0d0 / ((k * (k + 10.0d0)) / a)
    else if (k <= 1.75d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / (k + 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.35) {
		tmp = 1.0 / ((k * (k + 10.0)) / a);
	} else if (k <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.35:
		tmp = 1.0 / ((k * (k + 10.0)) / a)
	elif k <= 1.75:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / (k + 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.35)
		tmp = Float64(1.0 / Float64(Float64(k * Float64(k + 10.0)) / a));
	elseif (k <= 1.75)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / Float64(k + 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.35)
		tmp = 1.0 / ((k * (k + 10.0)) / a);
	elseif (k <= 1.75)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / (k + 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.35], N[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.75], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.3500000000000001

   1. Initial program 80.4%

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

      1. associate-*l/ 71.7%

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

      2. sqr-neg 71.7%

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

      3. associate-+l+ 71.7%

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

      4. sqr-neg 71.7%

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

      5. distribute-rgt-out 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in m around 0 44.8%

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

      1. clear-num 45.4%

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

      2. distribute-lft-in 45.4%

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

      3. associate-+l+ 45.4%

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

      4. associate-/r/ 44.7%

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

      5. associate-+l+ 44.7%

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

      6. distribute-lft-in 44.8%

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

      7. +-commutative 44.8%

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

      8. fma-define 44.8%

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

      9. +-commutative 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. associate-*l/ 44.8%

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

      2. associate-/l* 45.4%

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

   9. Applied egg-rr 45.4%

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

   10. Taylor expanded in k around inf 45.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{10 \cdot k + {k}^{2}}}{a}} \]
   11. Step-by-step derivation

       1. +-commutative 71.7%

          \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 71.7%

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

       3. distribute-rgt-in 71.7%

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

   12. Simplified 45.4%

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

   if -1.3500000000000001 < k < 1.75

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.2%

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

      1. *-commutative 37.2%

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

   8. Simplified 37.2%

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

   if 1.75 < k

   1. Initial program 84.8%

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

      1. associate-*l/ 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-+l+ 83.8%

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

      4. sqr-neg 83.8%

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

      5. distribute-rgt-out 83.8%

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

   3. Simplified 83.8%

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

      1. *-commutative 83.8%

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

      2. clear-num 83.7%

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

      3. un-div-inv 83.7%

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

      4. +-commutative 83.7%

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

      5. fma-define 83.7%

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

      6. +-commutative 83.7%

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

   6. Applied egg-rr 83.7%

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

   7. Taylor expanded in k around inf 82.8%

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

      1. +-commutative 82.8%

         \[\leadsto \frac{{k}^{m}}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 82.8%

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

      3. distribute-rgt-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in m around 0 61.6%

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

       1. associate-/r* 62.6%

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

       2. +-commutative 62.6%

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

   12. Simplified 62.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.35:\\ \;\;\;\;\frac{1}{\frac{k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.2e+37)
   (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a))
   (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.2e+37) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 3.2d+37) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.2e+37) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 3.2e+37:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.2e+37)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 3.2e+37)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.2e+37], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.20000000000000014e37

   1. Initial program 96.2%

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

      1. associate-*l/ 95.7%

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

      2. sqr-neg 95.7%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in m around 0 62.5%

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

      1. clear-num 62.6%

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

      2. distribute-lft-in 62.6%

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

      3. associate-+l+ 62.6%

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

      4. associate-/r/ 62.5%

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

      5. associate-+l+ 62.5%

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

      6. distribute-lft-in 62.5%

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

      7. +-commutative 62.5%

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

      8. fma-define 62.5%

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

      9. +-commutative 62.5%

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

   7. Applied egg-rr 62.5%

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

      1. associate-*l/ 62.5%

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

      2. associate-/l* 62.6%

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

   9. Applied egg-rr 62.6%

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

       1. fma-undefine 62.6%

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

   11. Applied egg-rr 62.6%

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

   if 3.20000000000000014e37 < m

   1. Initial program 74.2%

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

      1. associate-*l/ 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-+l+ 67.7%

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

      4. sqr-neg 67.7%

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

      5. distribute-rgt-out 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 11.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.55 \cdot 10^{+68} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2.55e+68) (not (<= k 0.1))) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.55e+68) || !(k <= 0.1)) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-2.55d+68)) .or. (.not. (k <= 0.1d0))) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.55e+68) || !(k <= 0.1)) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2.55e+68) or not (k <= 0.1):
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2.55e+68) || !(k <= 0.1))
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -2.55e+68) || ~((k <= 0.1)))
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2.55e+68], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -2.55e68 or 0.10000000000000001 < k

   1. Initial program 82.9%

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

      1. associate-*l/ 80.0%

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

      2. sqr-neg 80.0%

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

      3. associate-+l+ 80.0%

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

      4. sqr-neg 80.0%

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

      5. distribute-rgt-out 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in m around 0 58.9%

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

   6. Taylor expanded in k around 0 19.3%

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

      1. *-commutative 19.3%

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

   8. Simplified 19.3%

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

   9. Taylor expanded in k around inf 19.2%

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

   if -2.55e68 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-*l/ 99.1%

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

      2. sqr-neg 99.1%

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

      3. associate-+l+ 99.1%

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

      4. sqr-neg 99.1%

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

      5. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in m around 0 35.9%

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

   6. Taylor expanded in k around 0 35.3%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.55 \cdot 10^{+68} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2.4e+67) (* (/ a k) 0.1) (if (<= k 0.1) a (/ 0.1 (/ k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.4e+67) {
		tmp = (a / k) * 0.1;
	} else if (k <= 0.1) {
		tmp = a;
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-2.4d+67)) then
        tmp = (a / k) * 0.1d0
    else if (k <= 0.1d0) then
        tmp = a
    else
        tmp = 0.1d0 / (k / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.4e+67) {
		tmp = (a / k) * 0.1;
	} else if (k <= 0.1) {
		tmp = a;
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2.4e+67:
		tmp = (a / k) * 0.1
	elif k <= 0.1:
		tmp = a
	else:
		tmp = 0.1 / (k / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2.4e+67)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (k <= 0.1)
		tmp = a;
	else
		tmp = Float64(0.1 / Float64(k / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2.4e+67)
		tmp = (a / k) * 0.1;
	elseif (k <= 0.1)
		tmp = a;
	else
		tmp = 0.1 / (k / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2.4e+67], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[k, 0.1], a, N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.40000000000000002e67

   1. Initial program 76.9%

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

      1. associate-*l/ 69.2%

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

      2. sqr-neg 69.2%

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

      3. associate-+l+ 69.2%

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

      4. sqr-neg 69.2%

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

      5. distribute-rgt-out 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in m around 0 52.0%

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

   6. Taylor expanded in k around 0 16.3%

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

      1. *-commutative 16.3%

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

   8. Simplified 16.3%

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

   9. Taylor expanded in k around inf 16.3%

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

   if -2.40000000000000002e67 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-*l/ 99.1%

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

      2. sqr-neg 99.1%

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

      3. associate-+l+ 99.1%

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

      4. sqr-neg 99.1%

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

      5. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in m around 0 35.9%

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

   6. Taylor expanded in k around 0 35.3%

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

   if 0.10000000000000001 < k

   1. Initial program 85.3%

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

      1. associate-*l/ 84.3%

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

      2. sqr-neg 84.3%

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

      3. associate-+l+ 84.3%

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

      4. sqr-neg 84.3%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in m around 0 61.7%

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

   6. Taylor expanded in k around 0 20.5%

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

      1. *-commutative 20.5%

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

   8. Simplified 20.5%

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

   9. Taylor expanded in k around inf 20.4%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   10. Step-by-step derivation

       1. clear-num 20.8%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 20.8%

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

   11. Applied egg-rr 20.8%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.35e+42) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.35e+42) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.35d+42) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.35e+42) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.35e+42:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.35e+42)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.35e+42)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.35e+42], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.34999999999999993e42

   1. Initial program 96.2%

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

      1. associate-*l/ 95.7%

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

      2. sqr-neg 95.7%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in m around 0 62.5%

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

   if 2.34999999999999993e42 < m

   1. Initial program 74.2%

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

      1. associate-*l/ 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-+l+ 67.7%

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

      4. sqr-neg 67.7%

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

      5. distribute-rgt-out 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 11.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.08:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.08) (+ a (* -10.0 (* k a))) (/ 0.1 (/ k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.08) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.08d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = 0.1d0 / (k / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.08) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.08:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = 0.1 / (k / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.08)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(0.1 / Float64(k / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.08)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = 0.1 / (k / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.08], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0800000000000000017

   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-*l/ 91.8%

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

      2. sqr-neg 91.8%

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

      3. associate-+l+ 91.8%

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

      4. sqr-neg 91.8%

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

      5. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in m around 0 39.8%

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

   6. Taylor expanded in k around 0 31.0%

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

   if 0.0800000000000000017 < k

   1. Initial program 85.3%

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

      1. associate-*l/ 84.3%

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

      2. sqr-neg 84.3%

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

      3. associate-+l+ 84.3%

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

      4. sqr-neg 84.3%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in m around 0 61.7%

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

   6. Taylor expanded in k around 0 20.5%

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

      1. *-commutative 20.5%

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

   8. Simplified 20.5%

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

   9. Taylor expanded in k around inf 20.4%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   10. Step-by-step derivation

       1. clear-num 20.8%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 20.8%

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

   11. Applied egg-rr 20.8%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.08:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 20.8% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.9%

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

   1. associate-*l/ 88.9%

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

   2. sqr-neg 88.9%

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

   3. associate-+l+ 89.0%

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

   4. sqr-neg 89.0%

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

   5. distribute-rgt-out 89.0%

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

3. Simplified 89.0%

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

5. Taylor expanded in m around 0 48.1%

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

6. Taylor expanded in k around 0 18.8%

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

7. Final simplification 18.8%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))