Falkner and Boettcher, Appendix A

Percentage Accurate: 90.9% → 97.4%

Time: 11.0s

Alternatives: 12

Speedup: 1.0×

• 23.2% 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: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% 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 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in k around 0 99.8%

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

   if 0.10000000000000001 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

      1. *-commutative 77.2%

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

      2. clear-num 75.7%

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

      3. un-div-inv 75.7%

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

      4. +-commutative 75.7%

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

      5. fma-def 75.7%

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

      6. +-commutative 75.7%

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

   6. Applied egg-rr 75.7%

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

   7. Taylor expanded in k around inf 75.1%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   9. Simplified 75.1%

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

       1. associate-/l* 82.1%

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

       2. associate-/r/ 96.2%

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

   11. Applied egg-rr 96.2%

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

4. Final simplification 98.4%

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

Alternative 2: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -230000000 \lor \neg \left(m \leq 0.0005\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -230000000.0) (not (<= m 0.0005)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -230000000.0) || !(m <= 0.0005)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-230000000.0d0)) .or. (.not. (m <= 0.0005d0))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -230000000.0) || !(m <= 0.0005)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -230000000.0) or not (m <= 0.0005):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -230000000.0) || !(m <= 0.0005))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -230000000.0) || ~((m <= 0.0005)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -230000000.0], N[Not[LessEqual[m, 0.0005]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.3e8 or 5.0000000000000001e-4 < m

   1. Initial program 89.5%

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

      1. associate-*l/ 83.7%

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

      2. sqr-neg 83.7%

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

      3. associate-+l+ 83.7%

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

      4. sqr-neg 83.7%

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

      5. distribute-rgt-out 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.3e8 < m < 5.0000000000000001e-4

   1. Initial program 94.2%

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

      1. associate-*l/ 94.2%

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

      2. sqr-neg 94.2%

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

      3. associate-+l+ 94.2%

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

      4. sqr-neg 94.2%

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

      5. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in a around 0 94.2%

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

      1. +-commutative 94.2%

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

      2. distribute-lft-in 94.2%

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

      3. unpow2 94.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 94.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 94.2%

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

      6. distribute-lft-in 94.2%

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

      7. fma-udef 94.2%

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

      8. associate-*r/ 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in m around 0 93.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -230000000 \lor \neg \left(m \leq 0.0005\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in k around 0 99.8%

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

   if 1 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

      1. *-commutative 77.2%

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

      2. clear-num 75.7%

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

      3. un-div-inv 75.7%

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

      4. +-commutative 75.7%

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

      5. fma-def 75.7%

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

      6. +-commutative 75.7%

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

   6. Applied egg-rr 75.7%

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

   7. Taylor expanded in k around inf 75.1%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   9. Simplified 75.1%

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

       1. associate-/l* 82.1%

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

       2. associate-/r/ 96.2%

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

   11. Applied egg-rr 96.2%

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

   12. Taylor expanded in k around inf 95.5%

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

4. Final simplification 98.2%

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

Alternative 4: 44.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -6.2 \cdot 10^{+153} \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -6.2e+153) (not (<= k 0.075)))
   (/ a (* k (+ k 10.0)))
   (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -6.2e+153) || !(k <= 0.075)) {
		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 <= (-6.2d+153)) .or. (.not. (k <= 0.075d0))) 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 <= -6.2e+153) || !(k <= 0.075)) {
		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 <= -6.2e+153) or not (k <= 0.075):
		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 <= -6.2e+153) || !(k <= 0.075))
		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 <= -6.2e+153) || ~((k <= 0.075)))
		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, -6.2e+153], N[Not[LessEqual[k, 0.075]], $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 < -6.2e153 or 0.0749999999999999972 < k

   1. Initial program 80.0%

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

      1. associate-*l/ 75.6%

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

      2. sqr-neg 75.6%

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

      3. associate-+l+ 75.6%

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

      4. sqr-neg 75.6%

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

      5. distribute-rgt-out 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in m around 0 59.1%

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

      1. +-commutative 59.1%

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

      2. distribute-lft-in 59.1%

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

      3. unpow2 59.1%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 59.1%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 59.1%

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

      6. distribute-lft-in 59.1%

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

      7. fma-udef 59.1%

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

   7. Simplified 59.1%

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

   8. Taylor expanded in k around inf 58.6%

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

      1. +-commutative 73.8%

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

      2. unpow2 73.8%

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

      3. distribute-rgt-in 73.7%

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

   10. Simplified 58.6%

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

   if -6.2e153 < 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/ 96.5%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in m around 0 28.4%

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

      1. +-commutative 28.4%

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

      2. distribute-lft-in 28.4%

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

      3. unpow2 28.4%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 28.4%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 28.4%

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

      6. distribute-lft-in 28.4%

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

      7. fma-udef 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in k around 0 29.3%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.2 \cdot 10^{+153} \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -6.5e+153)
   (/ a (* k (+ k 10.0)))
   (if (<= k 0.075) (+ a (* -10.0 (* k a))) (/ (/ a k) (+ k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.5e+153) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-6.5d+153)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = (a / k) / (k + 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.5e+153) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -6.5e+153:
		tmp = a / (k * (k + 10.0))
	elif k <= 0.075:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = (a / k) / (k + 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -6.5e+153)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	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 <= -6.5e+153)
		tmp = a / (k * (k + 10.0));
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = (a / k) / (k + 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -6.5e+153], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -6.49999999999999972e153

   1. Initial program 66.7%

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

      1. associate-*l/ 66.7%

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

      2. sqr-neg 66.7%

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

      3. associate-+l+ 66.7%

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

      4. sqr-neg 66.7%

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

      5. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in m around 0 67.2%

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

      1. +-commutative 67.2%

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

      2. distribute-lft-in 67.2%

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

      3. unpow2 67.2%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 67.2%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 67.2%

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

      6. distribute-lft-in 67.2%

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

      7. fma-udef 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in k around inf 67.2%

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

      1. +-commutative 66.7%

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

      2. unpow2 66.7%

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

      3. distribute-rgt-in 66.7%

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

   10. Simplified 67.2%

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

   if -6.49999999999999972e153 < 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/ 96.5%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in m around 0 28.4%

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

      1. +-commutative 28.4%

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

      2. distribute-lft-in 28.4%

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

      3. unpow2 28.4%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 28.4%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 28.4%

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

      6. distribute-lft-in 28.4%

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

      7. fma-udef 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in k around 0 29.3%

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

   if 0.0749999999999999972 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in m around 0 57.6%

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

      1. +-commutative 57.6%

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

      2. distribute-lft-in 57.6%

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

      3. unpow2 57.6%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 57.6%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. fma-udef 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   10. Simplified 57.0%

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

   11. Taylor expanded in a around 0 57.0%

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

       1. associate-/r* 60.5%

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

       2. +-commutative 60.5%

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

   13. Simplified 60.5%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{+38}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 6.2e+38)
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))
   (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.2e+38) {
		tmp = a * (1.0 / (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 <= 6.2d+38) then
        tmp = a * (1.0d0 / (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 <= 6.2e+38) {
		tmp = a * (1.0 / (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 <= 6.2e+38:
		tmp = a * (1.0 / (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 <= 6.2e+38)
		tmp = Float64(a * Float64(1.0 / 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 <= 6.2e+38)
		tmp = a * (1.0 / (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, 6.2e+38], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $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 < 6.20000000000000035e38

   1. Initial program 96.6%

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

      1. associate-*l/ 95.4%

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

      2. sqr-neg 95.4%

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

      3. associate-+l+ 95.4%

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

      4. sqr-neg 95.4%

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

      5. distribute-rgt-out 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in a around 0 96.5%

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

      1. +-commutative 96.5%

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

      2. distribute-lft-in 96.6%

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

      3. unpow2 96.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 96.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 96.6%

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

      6. distribute-lft-in 96.5%

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

      7. fma-udef 96.6%

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

      8. associate-*r/ 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in m around 0 61.9%

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

   if 6.20000000000000035e38 < m

   1. Initial program 80.2%

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

      1. associate-*l/ 70.9%

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

      2. sqr-neg 70.9%

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

      3. associate-+l+ 70.9%

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

      4. sqr-neg 70.9%

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

      5. distribute-rgt-out 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. +-commutative 2.8%

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

      2. distribute-lft-in 2.8%

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

      3. unpow2 2.8%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 2.8%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 2.8%

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

      6. distribute-lft-in 2.8%

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

      7. fma-udef 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in k around 0 10.8%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{+38}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7.8e+31) (/ 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 <= 7.8e+31) {
		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 <= 7.8d+31) 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 <= 7.8e+31) {
		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 <= 7.8e+31:
		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 <= 7.8e+31)
		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 <= 7.8e+31)
		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, 7.8e+31], 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 < 7.79999999999999999e31

   1. Initial program 96.5%

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

      1. associate-*l/ 95.3%

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

      2. sqr-neg 95.3%

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

      3. associate-+l+ 95.3%

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

      4. sqr-neg 95.3%

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

      5. distribute-rgt-out 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in m around 0 62.9%

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

   if 7.79999999999999999e31 < m

   1. Initial program 80.9%

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

      1. associate-*l/ 71.9%

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

      2. sqr-neg 71.9%

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

      3. associate-+l+ 71.9%

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

      4. sqr-neg 71.9%

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

      5. distribute-rgt-out 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. distribute-lft-in 2.9%

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

      3. unpow2 2.9%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 2.9%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 2.9%

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

      6. distribute-lft-in 2.9%

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

      7. fma-udef 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in k around 0 10.6%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (+ a (* -10.0 (* k a))) (/ 0.1 (/ k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		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 <= 0.075d0) 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 <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		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 <= 0.075)
		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 <= 0.075)
		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, 0.075], 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 < 0.0749999999999999972

   1. Initial program 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in m around 0 32.7%

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

      1. +-commutative 32.7%

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

      2. distribute-lft-in 32.7%

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

      3. unpow2 32.7%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 32.7%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 32.7%

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

      6. distribute-lft-in 32.7%

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

      7. fma-udef 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in k around 0 27.5%

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

   if 0.0749999999999999972 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in m around 0 57.6%

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

      1. +-commutative 57.6%

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

      2. distribute-lft-in 57.6%

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

      3. unpow2 57.6%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 57.6%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. fma-udef 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   10. Simplified 57.0%

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

   11. Taylor expanded in k around 0 19.3%

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

       1. clear-num 20.6%

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

       2. un-div-inv 20.6%

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

   13. Applied egg-rr 20.6%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.5% accurate, 11.4× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = a;
	} else {
		tmp = (a / k) * 0.1;
	}
	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
    else
        tmp = (a / k) * 0.1d0
    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;
	} else {
		tmp = (a / k) * 0.1;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in m around 0 32.7%

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

      1. +-commutative 32.7%

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

      2. distribute-lft-in 32.7%

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

      3. unpow2 32.7%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 32.7%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 32.7%

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

      6. distribute-lft-in 32.7%

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

      7. fma-udef 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in k around 0 23.3%

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

   if 0.10000000000000001 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in m around 0 57.6%

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

      1. +-commutative 57.6%

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

      2. distribute-lft-in 57.6%

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

      3. unpow2 57.6%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 57.6%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. fma-udef 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   10. Simplified 57.0%

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

   11. Taylor expanded in k around 0 19.3%

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

4. Final simplification 21.8%

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

Alternative 10: 26.5% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.1d0) then
        tmp = a
    else
        tmp = a * (0.1d0 / k)
    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;
	} else {
		tmp = a * (0.1 / k);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in m around 0 32.7%

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

      1. +-commutative 32.7%

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

      2. distribute-lft-in 32.7%

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

      3. unpow2 32.7%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 32.7%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 32.7%

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

      6. distribute-lft-in 32.7%

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

      7. fma-udef 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in k around 0 23.3%

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

   if 0.10000000000000001 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in a around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. distribute-lft-in 82.4%

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

      3. unpow2 82.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 82.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 82.4%

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

      6. distribute-lft-in 82.4%

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

      7. fma-udef 82.4%

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

      8. associate-*r/ 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in m around 0 57.6%

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

   9. Taylor expanded in k around 0 19.3%

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

       1. *-commutative 19.3%

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

   11. Simplified 19.3%

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

   12. Taylor expanded in k around inf 19.3%

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

4. Final simplification 21.8%

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

Alternative 11: 26.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k 0.1) a (/ 0.1 (/ k a))))

C

double code(double a, double k, double m) {
	double tmp;
	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 <= 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 <= 0.1) {
		tmp = a;
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.1:
		tmp = a
	else:
		tmp = 0.1 / (k / a)
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 96.3%

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

      1. associate-*l/ 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-+l+ 93.1%

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

      4. sqr-neg 93.1%

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

      5. distribute-rgt-out 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in m around 0 32.7%

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

      1. +-commutative 32.7%

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

      2. distribute-lft-in 32.7%

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

      3. unpow2 32.7%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 32.7%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 32.7%

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

      6. distribute-lft-in 32.7%

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

      7. fma-udef 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in k around 0 23.3%

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

   if 0.10000000000000001 < k

   1. Initial program 82.4%

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

      1. associate-*l/ 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-+l+ 77.2%

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

      4. sqr-neg 77.2%

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

      5. distribute-rgt-out 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in m around 0 57.6%

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

      1. +-commutative 57.6%

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

      2. distribute-lft-in 57.6%

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

      3. unpow2 57.6%

         \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

      4. +-commutative 57.6%

         \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

      5. unpow2 57.6%

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

      6. distribute-lft-in 57.6%

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

      7. fma-udef 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-in 75.1%

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

   10. Simplified 57.0%

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

   11. Taylor expanded in k around 0 19.3%

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

       1. clear-num 20.6%

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

       2. un-div-inv 20.6%

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

   13. Applied egg-rr 20.6%

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

4. Final simplification 22.3%

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

Alternative 12: 20.1% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.1%

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

   1. associate-*l/ 87.2%

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

   2. sqr-neg 87.2%

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

   3. associate-+l+ 87.2%

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

   4. sqr-neg 87.2%

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

   5. distribute-rgt-out 87.2%

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

3. Simplified 87.2%

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

5. Taylor expanded in m around 0 42.1%

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

   1. +-commutative 42.1%

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

   2. distribute-lft-in 42.1%

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

   3. unpow2 42.1%

      \[\leadsto \frac{a}{\left(k \cdot 10 + \color{blue}{{k}^{2}}\right) + 1} \]

   4. +-commutative 42.1%

      \[\leadsto \frac{a}{\color{blue}{\left({k}^{2} + k \cdot 10\right)} + 1} \]

   5. unpow2 42.1%

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

   6. distribute-lft-in 42.1%

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

   7. fma-udef 42.1%

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

7. Simplified 42.1%

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

8. Taylor expanded in k around 0 16.1%

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

9. Final simplification 16.1%

   \[\leadsto a \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))