Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 98.0%

Time: 10.5s

Alternatives: 11

Speedup: 1.1×

• 23.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (* k k) (+ 1.0 (* k 10.0)))) 2e+296)
     (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+296) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = t_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 ** m)
    if ((t_0 / ((k * k) + (1.0d0 + (k * 10.0d0)))) <= 2d+296) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+296) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+296:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0)))) <= 2e+296)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+296)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+296], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 1.99999999999999996e296

   1. Initial program 99.5%

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

      1. associate-/l* 99.5%

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

      2. sqr-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. sqr-neg 99.5%

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

      5. distribute-rgt-out 99.5%

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

   3. Simplified 99.5%

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

   if 1.99999999999999996e296 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 64.0%

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

      1. associate-/l* 64.0%

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

      2. sqr-neg 64.0%

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

      3. associate-+l+ 64.0%

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

      4. sqr-neg 64.0%

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

      5. distribute-rgt-out 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot k}\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -2.5e-19)
     (/ t_0 (+ 1.0 (* k k)))
     (if (<= m 4.5e-31) (/ a (+ (* k k) (+ 1.0 (* k 10.0)))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -2.5e-19) {
		tmp = t_0 / (1.0 + (k * k));
	} else if (m <= 4.5e-31) {
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = t_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 ** m)
    if (m <= (-2.5d-19)) then
        tmp = t_0 / (1.0d0 + (k * k))
    else if (m <= 4.5d-31) then
        tmp = a / ((k * k) + (1.0d0 + (k * 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -2.5e-19) {
		tmp = t_0 / (1.0 + (k * k));
	} else if (m <= 4.5e-31) {
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -2.5e-19:
		tmp = t_0 / (1.0 + (k * k))
	elif m <= 4.5e-31:
		tmp = a / ((k * k) + (1.0 + (k * 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -2.5e-19)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * k)));
	elseif (m <= 4.5e-31)
		tmp = Float64(a / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -2.5e-19)
		tmp = t_0 / (1.0 + (k * k));
	elseif (m <= 4.5e-31)
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.5e-19], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.5e-31], N[(a / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.5000000000000002e-19

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.5000000000000002e-19 < m < 4.5000000000000004e-31

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in m around 0 98.7%

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

   if 4.5000000000000004e-31 < m

   1. Initial program 77.2%

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

      1. associate-/l* 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00125 \lor \neg \left(m \leq 4.5 \cdot 10^{-31}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.00125) (not (<= m 4.5e-31)))
   (* a (pow k m))
   (/ a (+ (* k k) (+ 1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00125) || !(m <= 4.5e-31)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / ((k * k) + (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 ((m <= (-0.00125d0)) .or. (.not. (m <= 4.5d-31))) then
        tmp = a * (k ** m)
    else
        tmp = a / ((k * k) + (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 ((m <= -0.00125) || !(m <= 4.5e-31)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.00125) or not (m <= 4.5e-31):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / ((k * k) + (1.0 + (k * 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.00125) || !(m <= 4.5e-31))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.00125) || ~((m <= 4.5e-31)))
		tmp = a * (k ^ m);
	else
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.00125], N[Not[LessEqual[m, 4.5e-31]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.00125000000000000003 or 4.5000000000000004e-31 < m

   1. Initial program 89.7%

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

      1. associate-/l* 89.7%

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

      2. sqr-neg 89.7%

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

      3. associate-+l+ 89.7%

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

      4. sqr-neg 89.7%

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

      5. distribute-rgt-out 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -0.00125000000000000003 < m < 4.5000000000000004e-31

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in m around 0 97.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00125 \lor \neg \left(m \leq 4.5 \cdot 10^{-31}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{if}\;k \leq -2.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{elif}\;k \leq 4.05 \cdot 10^{-283}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ a k) (/ 1.0 k))))
   (if (<= k -2.8e-299)
     (/ 1.0 (/ k (/ a k)))
     (if (<= k 4.05e-283)
       a
       (if (<= k 9.2e-272)
         t_0
         (if (<= k 2.8e-177)
           a
           (if (<= k 2.7e-102)
             t_0
             (if (<= k 0.075)
               (+ a (* -10.0 (* a k)))
               (/ a (* k (+ k 10.0)))))))))))

C

double code(double a, double k, double m) {
	double t_0 = (a / k) * (1.0 / k);
	double tmp;
	if (k <= -2.8e-299) {
		tmp = 1.0 / (k / (a / k));
	} else if (k <= 4.05e-283) {
		tmp = a;
	} else if (k <= 9.2e-272) {
		tmp = t_0;
	} else if (k <= 2.8e-177) {
		tmp = a;
	} else if (k <= 2.7e-102) {
		tmp = t_0;
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (a / k) * (1.0d0 / k)
    if (k <= (-2.8d-299)) then
        tmp = 1.0d0 / (k / (a / k))
    else if (k <= 4.05d-283) then
        tmp = a
    else if (k <= 9.2d-272) then
        tmp = t_0
    else if (k <= 2.8d-177) then
        tmp = a
    else if (k <= 2.7d-102) then
        tmp = t_0
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (a * k))
    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 t_0 = (a / k) * (1.0 / k);
	double tmp;
	if (k <= -2.8e-299) {
		tmp = 1.0 / (k / (a / k));
	} else if (k <= 4.05e-283) {
		tmp = a;
	} else if (k <= 9.2e-272) {
		tmp = t_0;
	} else if (k <= 2.8e-177) {
		tmp = a;
	} else if (k <= 2.7e-102) {
		tmp = t_0;
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (a / k) * (1.0 / k)
	tmp = 0
	if k <= -2.8e-299:
		tmp = 1.0 / (k / (a / k))
	elif k <= 4.05e-283:
		tmp = a
	elif k <= 9.2e-272:
		tmp = t_0
	elif k <= 2.8e-177:
		tmp = a
	elif k <= 2.7e-102:
		tmp = t_0
	elif k <= 0.075:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a / k) * Float64(1.0 / k))
	tmp = 0.0
	if (k <= -2.8e-299)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	elseif (k <= 4.05e-283)
		tmp = a;
	elseif (k <= 9.2e-272)
		tmp = t_0;
	elseif (k <= 2.8e-177)
		tmp = a;
	elseif (k <= 2.7e-102)
		tmp = t_0;
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (a / k) * (1.0 / k);
	tmp = 0.0;
	if (k <= -2.8e-299)
		tmp = 1.0 / (k / (a / k));
	elseif (k <= 4.05e-283)
		tmp = a;
	elseif (k <= 9.2e-272)
		tmp = t_0;
	elseif (k <= 2.8e-177)
		tmp = a;
	elseif (k <= 2.7e-102)
		tmp = t_0;
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.8e-299], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.05e-283], a, If[LessEqual[k, 9.2e-272], t$95$0, If[LessEqual[k, 2.8e-177], a, If[LessEqual[k, 2.7e-102], t$95$0, If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -2.8000000000000001e-299

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-/l* 81.5%

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

      3. sqr-neg 81.5%

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

      4. associate-+l+ 81.5%

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

      5. +-commutative 81.5%

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

      6. sqr-neg 81.5%

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

      7. distribute-rgt-out 81.5%

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

      8. fma-def 81.5%

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

      9. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in k around inf 52.3%

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

   6. Taylor expanded in m around 0 15.2%

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

      1. unpow2 15.2%

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

      2. associate-*r/ 9.4%

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

      3. distribute-rgt-out 9.6%

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

      4. +-commutative 9.6%

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

   8. Simplified 9.6%

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

      1. inv-pow 9.6%

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

      2. *-commutative 9.6%

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

   10. Applied egg-rr 9.6%

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

       1. unpow-1 9.6%

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

       2. *-commutative 9.6%

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

       3. associate-/r/ 9.6%

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

   12. Simplified 9.6%

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

   13. Taylor expanded in k around inf 19.3%

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

   if -2.8000000000000001e-299 < k < 4.04999999999999983e-283 or 9.19999999999999955e-272 < k < 2.79999999999999987e-177

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   6. Taylor expanded in m around 0 47.1%

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

   if 4.04999999999999983e-283 < k < 9.19999999999999955e-272 or 2.79999999999999987e-177 < k < 2.7e-102

   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. *-commutative 100.0%

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

      2. associate-/l* 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. fma-def 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in k around inf 82.2%

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

   6. Taylor expanded in m around 0 25.9%

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

      1. unpow2 25.9%

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

      2. associate-*r/ 25.9%

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

      3. distribute-rgt-out 25.9%

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

      4. +-commutative 25.9%

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

   8. Simplified 25.9%

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

      1. associate-*l/ 25.9%

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

      2. associate-/l* 25.9%

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

      3. times-frac 25.9%

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

   10. Applied egg-rr 25.9%

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

   11. Taylor expanded in k around inf 56.9%

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

   if 2.7e-102 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 59.7%

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

   6. Taylor expanded in k around 0 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   if 0.0749999999999999972 < k

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      2. associate-/l* 87.2%

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

      3. sqr-neg 87.2%

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

      4. associate-+l+ 87.2%

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

      5. +-commutative 87.2%

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

      6. sqr-neg 87.2%

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

      7. distribute-rgt-out 87.2%

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

      8. fma-def 87.2%

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

      9. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in k around inf 84.7%

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

   6. Taylor expanded in a around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. distribute-rgt-in 84.7%

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

   8. Simplified 84.7%

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

   9. Taylor expanded in m around 0 62.3%

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

       1. associate-/r* 56.8%

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

       2. +-commutative 56.8%

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

       3. associate-/l/ 62.3%

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

       4. *-commutative 62.3%

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

   11. Simplified 62.3%

       \[\leadsto \color{blue}{\frac{a}{k \cdot \left(k + 10\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{elif}\;k \leq 4.05 \cdot 10^{-283}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.45e-19)
   (/ 1.0 (/ k (/ a k)))
   (/ a (+ (* k k) (+ 1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.45e-19) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = a / ((k * k) + (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 (m <= (-1.45d-19)) then
        tmp = 1.0d0 / (k / (a / k))
    else
        tmp = a / ((k * k) + (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 (m <= -1.45e-19) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.45e-19:
		tmp = 1.0 / (k / (a / k))
	else:
		tmp = a / ((k * k) + (1.0 + (k * 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.45e-19)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	else
		tmp = Float64(a / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.45e-19)
		tmp = 1.0 / (k / (a / k));
	else
		tmp = a / ((k * k) + (1.0 + (k * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.45e-19], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.45e-19

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.1%

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

      3. sqr-neg 99.1%

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

      4. associate-+l+ 99.1%

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

      5. +-commutative 99.1%

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

      6. sqr-neg 99.1%

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

      7. distribute-rgt-out 99.1%

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

      8. fma-def 99.1%

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

      9. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around inf 84.1%

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

   6. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

      2. associate-*r/ 32.2%

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

      3. distribute-rgt-out 32.2%

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

      4. +-commutative 32.2%

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

   8. Simplified 32.2%

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

      1. inv-pow 32.2%

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

      2. *-commutative 32.2%

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

   10. Applied egg-rr 32.2%

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

       1. unpow-1 32.2%

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

       2. *-commutative 32.2%

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

       3. associate-/r/ 32.2%

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

   12. Simplified 32.2%

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

   13. Taylor expanded in k around inf 50.8%

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

   if -1.45e-19 < m

   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. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in m around 0 52.5%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.45e-19) (/ 1.0 (/ k (/ a k))) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.45e-19) {
		tmp = 1.0 / (k / (a / k));
	} 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 <= (-1.45d-19)) then
        tmp = 1.0d0 / (k / (a / k))
    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 <= -1.45e-19) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.45e-19:
		tmp = 1.0 / (k / (a / k))
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.45e-19)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	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 <= -1.45e-19)
		tmp = 1.0 / (k / (a / k));
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.45e-19], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $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 < -1.45e-19

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.1%

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

      3. sqr-neg 99.1%

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

      4. associate-+l+ 99.1%

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

      5. +-commutative 99.1%

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

      6. sqr-neg 99.1%

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

      7. distribute-rgt-out 99.1%

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

      8. fma-def 99.1%

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

      9. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around inf 84.1%

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

   6. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

      2. associate-*r/ 32.2%

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

      3. distribute-rgt-out 32.2%

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

      4. +-commutative 32.2%

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

   8. Simplified 32.2%

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

      1. inv-pow 32.2%

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

      2. *-commutative 32.2%

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

   10. Applied egg-rr 32.2%

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

       1. unpow-1 32.2%

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

       2. *-commutative 32.2%

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

       3. associate-/r/ 32.2%

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

   12. Simplified 32.2%

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

   13. Taylor expanded in k around inf 50.8%

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

   if -1.45e-19 < m

   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-/l* 88.4%

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

      2. sqr-neg 88.4%

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

      3. associate-+l+ 88.4%

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

      4. sqr-neg 88.4%

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

      5. distribute-rgt-out 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in m around 0 52.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-148}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.45e-148) (* (/ a k) (/ 1.0 k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.45e-148) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.45d-148)) then
        tmp = (a / k) * (1.0d0 / k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.45e-148) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.45e-148:
		tmp = (a / k) * (1.0 / k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.45e-148)
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.45e-148)
		tmp = (a / k) * (1.0 / k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.45e-148], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -1.4499999999999999e-148

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/l* 98.6%

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

      3. sqr-neg 98.6%

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

      4. associate-+l+ 98.6%

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

      5. +-commutative 98.6%

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

      6. sqr-neg 98.6%

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

      7. distribute-rgt-out 98.6%

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

      8. fma-def 98.6%

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

      9. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in k around inf 80.4%

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

   6. Taylor expanded in m around 0 43.2%

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

      1. unpow2 43.2%

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

      2. associate-*r/ 36.0%

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

      3. distribute-rgt-out 36.0%

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

      4. +-commutative 36.0%

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

   8. Simplified 36.0%

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

      1. associate-*l/ 43.2%

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

      2. associate-/l* 43.7%

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

      3. times-frac 36.1%

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

   10. Applied egg-rr 36.1%

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

   11. Taylor expanded in k around inf 51.5%

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

   if -1.4499999999999999e-148 < m

   1. Initial program 87.0%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. associate-+l+ 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-rgt-out 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in k around 0 82.0%

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

   6. Taylor expanded in m around 0 29.2%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-148}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.65 \cdot 10^{-150}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.65e-150) (* (/ a k) (/ 1.0 k)) (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.65e-150) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = a + (-10.0 * (a * 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 (m <= (-1.65d-150)) then
        tmp = (a / k) * (1.0d0 / k)
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.65e-150) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.65e-150:
		tmp = (a / k) * (1.0 / k)
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.65e-150)
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.65e-150)
		tmp = (a / k) * (1.0 / k);
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.65e-150], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.6500000000000001e-150

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/l* 98.6%

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

      3. sqr-neg 98.6%

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

      4. associate-+l+ 98.6%

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

      5. +-commutative 98.6%

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

      6. sqr-neg 98.6%

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

      7. distribute-rgt-out 98.6%

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

      8. fma-def 98.6%

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

      9. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in k around inf 80.4%

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

   6. Taylor expanded in m around 0 43.2%

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

      1. unpow2 43.2%

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

      2. associate-*r/ 36.0%

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

      3. distribute-rgt-out 36.0%

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

      4. +-commutative 36.0%

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

   8. Simplified 36.0%

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

      1. associate-*l/ 43.2%

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

      2. associate-/l* 43.7%

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

      3. times-frac 36.1%

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

   10. Applied egg-rr 36.1%

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

   11. Taylor expanded in k around inf 51.5%

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

   if -1.6500000000000001e-150 < m

   1. Initial program 87.0%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. associate-+l+ 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-rgt-out 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in m around 0 46.7%

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

   6. Taylor expanded in k around 0 30.1%

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

      1. *-commutative 30.1%

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

   8. Simplified 30.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.65 \cdot 10^{-150}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e-19) (/ 1.0 (/ k (/ a k))) (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e-19) {
		tmp = 1.0 / (k / (a / k));
	} 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 (m <= (-1.4d-19)) then
        tmp = 1.0d0 / (k / (a / k))
    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 (m <= -1.4e-19) {
		tmp = 1.0 / (k / (a / k));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.4e-19:
		tmp = 1.0 / (k / (a / k))
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e-19)
		tmp = Float64(1.0 / Float64(k / Float64(a / k)));
	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 (m <= -1.4e-19)
		tmp = 1.0 / (k / (a / k));
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e-19], N[(1.0 / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.40000000000000001e-19

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.1%

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

      3. sqr-neg 99.1%

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

      4. associate-+l+ 99.1%

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

      5. +-commutative 99.1%

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

      6. sqr-neg 99.1%

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

      7. distribute-rgt-out 99.1%

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

      8. fma-def 99.1%

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

      9. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around inf 84.1%

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

   6. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

      2. associate-*r/ 32.2%

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

      3. distribute-rgt-out 32.2%

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

      4. +-commutative 32.2%

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

   8. Simplified 32.2%

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

      1. inv-pow 32.2%

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

      2. *-commutative 32.2%

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

   10. Applied egg-rr 32.2%

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

       1. unpow-1 32.2%

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

       2. *-commutative 32.2%

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

       3. associate-/r/ 32.2%

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

   12. Simplified 32.2%

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

   13. Taylor expanded in k around inf 50.8%

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

   if -1.40000000000000001e-19 < m

   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-/l* 88.4%

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

      2. sqr-neg 88.4%

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

      3. associate-+l+ 88.4%

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

      4. sqr-neg 88.4%

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

      5. distribute-rgt-out 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in m around 0 52.5%

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

   6. Taylor expanded in k around 0 36.1%

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

      1. *-commutative 36.1%

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

   8. Simplified 36.1%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m -1.6e-11) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.6e-11) {
		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 (m <= (-1.6d-11)) 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 (m <= -1.6e-11) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.6e-11:
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.6e-11)
		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 (m <= -1.6e-11)
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.6e-11], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -1.59999999999999997e-11

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 30.2%

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

   6. Taylor expanded in k around 0 11.8%

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

      1. *-commutative 11.8%

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

   8. Simplified 11.8%

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

   9. Taylor expanded in k around inf 22.4%

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

   if -1.59999999999999997e-11 < m

   1. Initial program 88.0%

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

      1. associate-/l* 88.0%

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

      2. sqr-neg 88.0%

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

      3. associate-+l+ 88.0%

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

      4. sqr-neg 88.0%

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

      5. distribute-rgt-out 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in k around 0 76.6%

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

   6. Taylor expanded in m around 0 30.3%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 20.9% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 92.6%

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

   1. associate-/l* 92.6%

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

   2. sqr-neg 92.6%

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

   3. associate-+l+ 92.6%

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

   4. sqr-neg 92.6%

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

   5. distribute-rgt-out 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in k around 0 84.7%

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

6. Taylor expanded in m around 0 20.1%

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

7. Final simplification 20.1%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))