Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 98.8%

Time: 12.9s

Alternatives: 13

Speedup: 1.0×

• 21.8% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -3.15 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;m \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -3.15e-80)
     (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))
     (if (<= m 1e-17)
       (/ 1.0 (+ (/ 1.0 a) (* k (+ (* 10.0 (/ 1.0 a)) (/ k a)))))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -3.15e-80) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else if (m <= 1e-17) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} 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 <= (-3.15d-80)) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else if (m <= 1d-17) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((10.0d0 * (1.0d0 / a)) + (k / a))))
    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 <= -3.15e-80) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else if (m <= 1e-17) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -3.15e-80:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	elif m <= 1e-17:
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -3.15e-80)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	elseif (m <= 1e-17)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(10.0 * Float64(1.0 / a)) + Float64(k / a)))));
	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 <= -3.15e-80)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	elseif (m <= 1e-17)
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	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, -3.15e-80], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1e-17], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(10.0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.14999999999999983e-80

   1. Initial program 100.0%

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

   if -3.14999999999999983e-80 < m < 1.00000000000000007e-17

   1. Initial program 92.0%

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

      1. associate-/l* 91.9%

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

      2. remove-double-neg 91.9%

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

      3. distribute-frac-neg2 91.9%

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

      4. distribute-neg-frac2 91.9%

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

      5. remove-double-neg 91.9%

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

      6. sqr-neg 91.9%

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

      7. associate-+l+ 91.9%

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

      8. sqr-neg 91.9%

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

      9. distribute-rgt-out 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in m around 0 91.9%

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

      1. associate-*r/ 92.0%

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

      2. clear-num 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

      5. +-commutative 91.8%

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

      6. *-rgt-identity 91.8%

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

   7. Applied egg-rr 91.8%

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

   8. Taylor expanded in k around 0 99.3%

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

   if 1.00000000000000007e-17 < m

   1. Initial program 78.9%

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

      1. associate-/l* 78.9%

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

      2. remove-double-neg 78.9%

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

      3. distribute-frac-neg2 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. remove-double-neg 78.9%

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

      6. sqr-neg 78.9%

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

      7. associate-+l+ 78.9%

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

      8. sqr-neg 78.9%

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

      9. distribute-rgt-out 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in k around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.15 \cdot 10^{-80}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;m \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-81}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.02e-81)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m 1e-17)
     (/ 1.0 (+ (/ 1.0 a) (* k (+ (* 10.0 (/ 1.0 a)) (/ k a)))))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02e-81) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 1e-17) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.02d-81)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else if (m <= 1d-17) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((10.0d0 * (1.0d0 / a)) + (k / a))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02e-81) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 1e-17) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.02e-81:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	elif m <= 1e-17:
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.02e-81)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	elseif (m <= 1e-17)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(10.0 * Float64(1.0 / a)) + Float64(k / a)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.02e-81)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	elseif (m <= 1e-17)
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.02e-81], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1e-17], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(10.0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.01999999999999998e-81

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -1.01999999999999998e-81 < m < 1.00000000000000007e-17

   1. Initial program 92.0%

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

      1. associate-/l* 91.9%

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

      2. remove-double-neg 91.9%

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

      3. distribute-frac-neg2 91.9%

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

      4. distribute-neg-frac2 91.9%

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

      5. remove-double-neg 91.9%

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

      6. sqr-neg 91.9%

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

      7. associate-+l+ 91.9%

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

      8. sqr-neg 91.9%

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

      9. distribute-rgt-out 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in m around 0 91.9%

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

      1. associate-*r/ 92.0%

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

      2. clear-num 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

      5. +-commutative 91.8%

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

      6. *-rgt-identity 91.8%

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

   7. Applied egg-rr 91.8%

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

   8. Taylor expanded in k around 0 99.3%

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

   if 1.00000000000000007e-17 < m

   1. Initial program 78.9%

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

      1. associate-/l* 78.9%

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

      2. remove-double-neg 78.9%

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

      3. distribute-frac-neg2 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. remove-double-neg 78.9%

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

      6. sqr-neg 78.9%

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

      7. associate-+l+ 78.9%

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

      8. sqr-neg 78.9%

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

      9. distribute-rgt-out 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in k around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-81}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2100000000000 \lor \neg \left(m \leq 10^{-17}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2100000000000.0) (not (<= m 1e-17)))
   (* a (pow k m))
   (/ 1.0 (+ (/ 1.0 a) (* k (+ (* 10.0 (/ 1.0 a)) (/ k a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2100000000000.0) || !(m <= 1e-17)) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (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 <= (-2100000000000.0d0)) .or. (.not. (m <= 1d-17))) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((10.0d0 * (1.0d0 / a)) + (k / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2100000000000.0) || !(m <= 1e-17)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -2100000000000.0) or not (m <= 1e-17):
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2100000000000.0) || !(m <= 1e-17))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(10.0 * Float64(1.0 / a)) + Float64(k / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -2100000000000.0) || ~((m <= 1e-17)))
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2100000000000.0], N[Not[LessEqual[m, 1e-17]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(10.0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.1e12 or 1.00000000000000007e-17 < m

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. remove-double-neg 89.9%

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

      3. distribute-frac-neg2 89.9%

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

      4. distribute-neg-frac2 89.9%

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

      5. remove-double-neg 89.9%

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

      6. sqr-neg 89.9%

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

      7. associate-+l+ 89.9%

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

      8. sqr-neg 89.9%

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

      9. distribute-rgt-out 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in k around 0 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   if -2.1e12 < m < 1.00000000000000007e-17

   1. Initial program 92.9%

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

      1. associate-/l* 92.8%

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

      2. remove-double-neg 92.8%

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

      3. distribute-frac-neg2 92.8%

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

      4. distribute-neg-frac2 92.8%

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

      5. remove-double-neg 92.8%

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

      6. sqr-neg 92.8%

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

      7. associate-+l+ 92.8%

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

      8. sqr-neg 92.8%

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

      9. distribute-rgt-out 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in m around 0 92.8%

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

      1. associate-*r/ 92.9%

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

      2. clear-num 92.7%

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

      3. +-commutative 92.7%

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

      4. fma-define 92.7%

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

      5. +-commutative 92.7%

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

      6. *-rgt-identity 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in k around 0 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2100000000000 \lor \neg \left(m \leq 10^{-17}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 78:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 78.0)
   (/ 1.0 (+ (/ 1.0 a) (* k (+ (* 10.0 (/ 1.0 a)) (/ k a)))))
   (* a (+ 1.0 (* k (- (* k 99.0) 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 78.0) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 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 <= 78.0d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((10.0d0 * (1.0d0 / a)) + (k / a))))
    else
        tmp = a * (1.0d0 + (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 78.0) {
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 78.0:
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))))
	else:
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 78.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(10.0 * Float64(1.0 / a)) + Float64(k / a)))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 78.0)
		tmp = 1.0 / ((1.0 / a) + (k * ((10.0 * (1.0 / a)) + (k / a))));
	else
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 78.0], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(10.0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 78

   1. Initial program 96.2%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 66.1%

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

      1. associate-*r/ 66.2%

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

      2. clear-num 66.4%

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

      3. +-commutative 66.4%

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

      4. fma-define 66.4%

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

      5. +-commutative 66.4%

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

      6. *-rgt-identity 66.4%

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

   7. Applied egg-rr 66.4%

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

   8. Taylor expanded in k around 0 67.9%

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

   if 78 < m

   1. Initial program 78.1%

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

      1. associate-/l* 78.1%

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

      2. remove-double-neg 78.1%

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

      3. distribute-frac-neg2 78.1%

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

      4. distribute-neg-frac2 78.1%

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

      5. remove-double-neg 78.1%

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

      6. sqr-neg 78.1%

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

      7. associate-+l+ 78.1%

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

      8. sqr-neg 78.1%

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

      9. distribute-rgt-out 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 29.3%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 78:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{+83} \lor \neg \left(k \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2.1e+83) (not (<= k 3.5e-7)))
   (* 0.1 (/ a k))
   (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.1e+83) || !(k <= 3.5e-7)) {
		tmp = 0.1 * (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 ((k <= (-2.1d+83)) .or. (.not. (k <= 3.5d-7))) then
        tmp = 0.1d0 * (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 ((k <= -2.1e+83) || !(k <= 3.5e-7)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2.1e+83) or not (k <= 3.5e-7):
		tmp = 0.1 * (a / k)
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2.1e+83) || !(k <= 3.5e-7))
		tmp = Float64(0.1 * 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 ((k <= -2.1e+83) || ~((k <= 3.5e-7)))
		tmp = 0.1 * (a / k);
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2.1e+83], N[Not[LessEqual[k, 3.5e-7]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -2.10000000000000002e83 or 3.49999999999999984e-7 < k

   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. *-commutative 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in m around 0 60.8%

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

   6. Taylor expanded in k around 0 21.5%

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

      1. *-commutative 21.5%

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

   8. Simplified 21.5%

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

   9. Taylor expanded in k around inf 21.5%

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

   if -2.10000000000000002e83 < k < 3.49999999999999984e-7

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.8%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{+83} \lor \neg \left(k \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (* a (+ 1.0 (* k (- (* k 99.0) 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.1d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (1.0d0 + (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.1:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.1)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 96.2%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 66.5%

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

   if 2.10000000000000009 < m

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-neg-frac2 78.4%

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

      5. remove-double-neg 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-+l+ 78.4%

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

      8. sqr-neg 78.4%

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

      9. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 28.9%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.0)
   (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a))
   (* a (+ 1.0 (* k (- (* k 99.0) 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.0d0) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = a * (1.0d0 + (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.0:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.0)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.0], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2

   1. Initial program 96.2%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. clear-num 66.8%

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

      3. +-commutative 66.8%

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

      4. fma-define 66.8%

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

      5. +-commutative 66.8%

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

      6. *-rgt-identity 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in a around 0 66.8%

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

   if 2 < m

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-neg-frac2 78.4%

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

      5. remove-double-neg 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-+l+ 78.4%

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

      8. sqr-neg 78.4%

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

      9. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 28.9%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+80} \lor \neg \left(k \leq 1.9 \cdot 10^{-7}\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2.7e+80) (not (<= k 1.9e-7))) (* 0.1 (/ a k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.7e+80) || !(k <= 1.9e-7)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-2.7d+80)) .or. (.not. (k <= 1.9d-7))) then
        tmp = 0.1d0 * (a / k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.7e+80) || !(k <= 1.9e-7)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2.7e+80) or not (k <= 1.9e-7):
		tmp = 0.1 * (a / k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2.7e+80) || !(k <= 1.9e-7))
		tmp = Float64(0.1 * Float64(a / k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -2.7e+80) || ~((k <= 1.9e-7)))
		tmp = 0.1 * (a / k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2.7e+80], N[Not[LessEqual[k, 1.9e-7]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -2.69999999999999983e80 or 1.90000000000000007e-7 < k

   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. *-commutative 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in m around 0 60.8%

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

   6. Taylor expanded in k around 0 21.5%

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

      1. *-commutative 21.5%

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

   8. Simplified 21.5%

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

   9. Taylor expanded in k around inf 21.5%

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

   if -2.69999999999999983e80 < k < 1.90000000000000007e-7

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.8%

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

   6. Taylor expanded in k around 0 37.8%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+80} \lor \neg \left(k \leq 1.9 \cdot 10^{-7}\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.2) (/ a (+ 1.0 (* k 10.0))) (+ a (* k (* 99.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (99.0 * (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 <= 2.2d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a + (k * (99.0d0 * (a * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (99.0 * (a * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.2:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a + (k * (99.0 * (a * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a + Float64(k * Float64(99.0 * Float64(a * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a + (k * (99.0 * (a * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.2], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(99.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in m around 0 66.5%

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

   6. Taylor expanded in k around 0 41.4%

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

      1. *-commutative 41.4%

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

   8. Simplified 41.4%

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

   if 2.2000000000000002 < m

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-neg-frac2 78.4%

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

      5. remove-double-neg 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-+l+ 78.4%

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

      8. sqr-neg 78.4%

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

      9. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 26.4%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. mul-1-neg 26.4%

         \[\leadsto a + k \cdot \left(\color{blue}{\left(-k \cdot \left(a + -100 \cdot a\right)\right)} + \left(-10\right) \cdot a\right) \]

      3. distribute-rgt1-in 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + \left(-10\right) \cdot a\right) \]

      4. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(\color{blue}{-99} \cdot a\right)\right) + \left(-10\right) \cdot a\right) \]

      5. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      6. *-commutative 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

   8. Simplified 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + a \cdot -10\right)} \]

   9. Taylor expanded in k around inf 26.4%

      \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.2) (/ a (+ 1.0 (* k 10.0))) (+ a (* k (* k (* a 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (k * (a * 99.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.2d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a + (k * (k * (a * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (k * (a * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.2:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a + (k * (k * (a * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a + Float64(k * Float64(k * Float64(a * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a + (k * (k * (a * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.2], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in m around 0 66.5%

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

   6. Taylor expanded in k around 0 41.4%

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

      1. *-commutative 41.4%

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

   8. Simplified 41.4%

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

   if 2.2000000000000002 < m

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-neg-frac2 78.4%

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

      5. remove-double-neg 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-+l+ 78.4%

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

      8. sqr-neg 78.4%

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

      9. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 26.4%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. mul-1-neg 26.4%

         \[\leadsto a + k \cdot \left(\color{blue}{\left(-k \cdot \left(a + -100 \cdot a\right)\right)} + \left(-10\right) \cdot a\right) \]

      3. distribute-rgt1-in 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + \left(-10\right) \cdot a\right) \]

      4. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(\color{blue}{-99} \cdot a\right)\right) + \left(-10\right) \cdot a\right) \]

      5. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      6. *-commutative 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

   8. Simplified 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + a \cdot -10\right)} \]

   9. Taylor expanded in k around inf 26.4%

      \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 26.4%

          \[\leadsto a + k \cdot \color{blue}{\left(\left(99 \cdot a\right) \cdot k\right)} \]

       2. *-commutative 26.4%

          \[\leadsto a + k \cdot \left(\color{blue}{\left(a \cdot 99\right)} \cdot k\right) \]

       3. metadata-eval 26.4%

          \[\leadsto a + k \cdot \left(\left(a \cdot \color{blue}{\left(--99\right)}\right) \cdot k\right) \]

       4. distribute-rgt-neg-in 26.4%

          \[\leadsto a + k \cdot \left(\color{blue}{\left(-a \cdot -99\right)} \cdot k\right) \]

       5. *-commutative 26.4%

          \[\leadsto a + k \cdot \color{blue}{\left(k \cdot \left(-a \cdot -99\right)\right)} \]

       6. distribute-rgt-neg-in 26.4%

          \[\leadsto a + k \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(--99\right)\right)}\right) \]

       7. metadata-eval 26.4%

          \[\leadsto a + k \cdot \left(k \cdot \left(a \cdot \color{blue}{99}\right)\right) \]

   11. Simplified 26.4%

       \[\leadsto a + k \cdot \color{blue}{\left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* k (* k (* a 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (k * (a * 99.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (k * (a * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (k * (a * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (k * (a * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(k * Float64(a * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (k * (a * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2

   1. Initial program 96.2%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 66.5%

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

   if 2 < m

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. remove-double-neg 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-neg-frac2 78.4%

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

      5. remove-double-neg 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-+l+ 78.4%

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

      8. sqr-neg 78.4%

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

      9. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in m around 0 3.3%

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

   6. Taylor expanded in k around 0 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 26.4%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. mul-1-neg 26.4%

         \[\leadsto a + k \cdot \left(\color{blue}{\left(-k \cdot \left(a + -100 \cdot a\right)\right)} + \left(-10\right) \cdot a\right) \]

      3. distribute-rgt1-in 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + \left(-10\right) \cdot a\right) \]

      4. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(\color{blue}{-99} \cdot a\right)\right) + \left(-10\right) \cdot a\right) \]

      5. metadata-eval 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      6. *-commutative 26.4%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

   8. Simplified 26.4%

      \[\leadsto \color{blue}{a + k \cdot \left(\left(-k \cdot \left(-99 \cdot a\right)\right) + a \cdot -10\right)} \]

   9. Taylor expanded in k around inf 26.4%

      \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 26.4%

          \[\leadsto a + k \cdot \color{blue}{\left(\left(99 \cdot a\right) \cdot k\right)} \]

       2. *-commutative 26.4%

          \[\leadsto a + k \cdot \left(\color{blue}{\left(a \cdot 99\right)} \cdot k\right) \]

       3. metadata-eval 26.4%

          \[\leadsto a + k \cdot \left(\left(a \cdot \color{blue}{\left(--99\right)}\right) \cdot k\right) \]

       4. distribute-rgt-neg-in 26.4%

          \[\leadsto a + k \cdot \left(\color{blue}{\left(-a \cdot -99\right)} \cdot k\right) \]

       5. *-commutative 26.4%

          \[\leadsto a + k \cdot \color{blue}{\left(k \cdot \left(-a \cdot -99\right)\right)} \]

       6. distribute-rgt-neg-in 26.4%

          \[\leadsto a + k \cdot \left(k \cdot \color{blue}{\left(a \cdot \left(--99\right)\right)}\right) \]

       7. metadata-eval 26.4%

          \[\leadsto a + k \cdot \left(k \cdot \left(a \cdot \color{blue}{99}\right)\right) \]

   11. Simplified 26.4%

       \[\leadsto a + k \cdot \color{blue}{\left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.4% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1 + k \cdot 10} \end{array} \]

FPCore

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

C

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

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 / (1.0d0 + (k * 10.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. *-commutative 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in m around 0 48.2%

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

6. Taylor expanded in k around 0 30.4%

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

   1. *-commutative 30.4%

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

8. Simplified 30.4%

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

9. Final simplification 30.4%

   \[\leadsto \frac{a}{1 + k \cdot 10} \]
10. Add Preprocessing

Alternative 13: 20.0% 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 91.0%

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

   1. associate-/l* 91.0%

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

   2. remove-double-neg 91.0%

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

   3. distribute-frac-neg2 91.0%

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

   4. distribute-neg-frac2 91.0%

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

   5. remove-double-neg 91.0%

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

   6. sqr-neg 91.0%

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

   7. associate-+l+ 91.0%

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

   8. sqr-neg 91.0%

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

   9. distribute-rgt-out 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in m around 0 48.2%

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

6. Taylor expanded in k around 0 22.8%

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

7. Final simplification 22.8%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))