Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 97.1%

Time: 12.6s

Alternatives: 15

Speedup: 1.0×

• 22.2% of points produce a very large (infinite) output. You may want to add a precondition.

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: 90.4% 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: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -4.5e-14)
     t_0
     (if (<= m 1.65e-10) (/ a (+ 1.0 (* k (+ k 10.0)))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -4.5e-14) {
		tmp = t_0;
	} else if (m <= 1.65e-10) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-4.5d-14)) then
        tmp = t_0
    else if (m <= 1.65d-10) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -4.5e-14) {
		tmp = t_0;
	} else if (m <= 1.65e-10) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -4.5e-14:
		tmp = t_0
	elif m <= 1.65e-10:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -4.5e-14)
		tmp = t_0;
	elseif (m <= 1.65e-10)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -4.5e-14)
		tmp = t_0;
	elseif (m <= 1.65e-10)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.5e-14], t$95$0, If[LessEqual[m, 1.65e-10], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -4.4999999999999998e-14 or 1.65e-10 < m

   1. Initial program 87.8%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   5. Simplified 100.0%

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

   if -4.4999999999999998e-14 < m < 1.65e-10

   1. Initial program 89.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 89.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 5e+133) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+133) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 5d+133) then
        tmp = t_1
    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 t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+133) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 5e+133:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e+133)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 5e+133)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+133], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.99999999999999961e133

   1. Initial program 95.5%

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

   if 4.99999999999999961e133 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 58.0%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.32e-9)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.32e-9) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} 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 <= 1.32d-9) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    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 <= 1.32e-9) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.32e-9:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.32e-9)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.32e-9)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.32e-9], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.32e-9

   1. Initial program 94.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\right), \color{blue}{a}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({k}^{m}\right), \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right), a\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \left(\left(1 + 10 \cdot k\right) + k \cdot k\right)\right), a\right) \]

      6. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \left(1 + \left(10 \cdot k + k \cdot k\right)\right)\right), a\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \left(10 \cdot k + k \cdot k\right)\right)\right), a\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \left(k \cdot \left(10 + k\right)\right)\right)\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right), a\right) \]

      10. +-lowering-+.f64 94.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, m\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(10, k\right)\right)\right)\right), a\right) \]

   4. Applied egg-rr 94.5%

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

   if 1.32e-9 < m

   1. Initial program 75.6%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left({k}^{m}\right)}\right) \]

      2. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{pow.f64}\left(k, \color{blue}{m}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.38:\\ \;\;\;\;a \cdot \frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.19:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot \left(-1 + \frac{100}{k \cdot k}\right)\right)\right)}{10 - k}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.38)
   (* a (/ (+ 1.0 (/ (+ -10.0 (/ 100.0 k)) k)) (* k k)))
   (if (<= m 0.19)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (if (<= m 1.25e+218)
       (/
        a
        (+ 1.0 (/ (* k (* k (* k (+ -1.0 (/ 100.0 (* k k)))))) (- 10.0 k))))
       (+ a (* k (* a (+ 10.0 (* k 99.0)))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.38) {
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	} else if (m <= 0.19) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 1.25e+218) {
		tmp = a / (1.0 + ((k * (k * (k * (-1.0 + (100.0 / (k * k)))))) / (10.0 - k)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 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 <= (-0.38d0)) then
        tmp = a * ((1.0d0 + (((-10.0d0) + (100.0d0 / k)) / k)) / (k * k))
    else if (m <= 0.19d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= 1.25d+218) then
        tmp = a / (1.0d0 + ((k * (k * (k * ((-1.0d0) + (100.0d0 / (k * k)))))) / (10.0d0 - k)))
    else
        tmp = a + (k * (a * (10.0d0 + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.38) {
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	} else if (m <= 0.19) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 1.25e+218) {
		tmp = a / (1.0 + ((k * (k * (k * (-1.0 + (100.0 / (k * k)))))) / (10.0 - k)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.38:
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k))
	elif m <= 0.19:
		tmp = a / (1.0 + (k * (k + 10.0)))
	elif m <= 1.25e+218:
		tmp = a / (1.0 + ((k * (k * (k * (-1.0 + (100.0 / (k * k)))))) / (10.0 - k)))
	else:
		tmp = a + (k * (a * (10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.38)
		tmp = Float64(a * Float64(Float64(1.0 + Float64(Float64(-10.0 + Float64(100.0 / k)) / k)) / Float64(k * k)));
	elseif (m <= 0.19)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= 1.25e+218)
		tmp = Float64(a / Float64(1.0 + Float64(Float64(k * Float64(k * Float64(k * Float64(-1.0 + Float64(100.0 / Float64(k * k)))))) / Float64(10.0 - k))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.38)
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	elseif (m <= 0.19)
		tmp = a / (1.0 + (k * (k + 10.0)));
	elseif (m <= 1.25e+218)
		tmp = a / (1.0 + ((k * (k * (k * (-1.0 + (100.0 / (k * k)))))) / (10.0 - k)));
	else
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.38], N[(a * N[(N[(1.0 + N[(N[(-10.0 + N[(100.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.19], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.25e+218], N[(a / N[(1.0 + N[(N[(k * N[(k * N[(k * N[(-1.0 + N[(100.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(10.0 - k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.38

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\color{blue}{\left(10 \cdot k\right)}, \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 10\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right)\right) \]

      2. *-lowering-*.f64 43.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 10\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right)\right) \]

   8. Simplified 43.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{1}{k \cdot \frac{10 \cdot 10 - k \cdot k}{10 - k}} \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{k \cdot \frac{100 - k \cdot k}{10 - k}} \cdot a \]

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{10 - k}{\left(100 - k \cdot k\right) \cdot k}\right), \color{blue}{a}\right) \]

   10. Applied egg-rr 43.9%

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

   11. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{100}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}}{{k}^{2}}\right)}, a\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{100}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}\right), \left({k}^{2}\right)\right), a\right) \]

   13. Simplified 75.0%

       \[\leadsto \color{blue}{\frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}} \cdot a \]

   if -0.38 < m < 0.19

   1. Initial program 89.2%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 88.4%

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

   if 0.19 < m < 1.24999999999999996e218

   1. Initial program 81.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 3.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(10 + k\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{10 \cdot 10 - k \cdot k}{10 - k} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(10 \cdot 10 - k \cdot k\right) \cdot k}{\color{blue}{10 - k}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(10 \cdot 10 - k \cdot k\right) \cdot k\right), \color{blue}{\left(10 - k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(10 \cdot 10 - k \cdot k\right), k\right), \left(\color{blue}{10} - k\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(10 \cdot 10\right), \left(k \cdot k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \left(k \cdot k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      9. --lowering--.f64 3.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, k\right)\right), k\right), \mathsf{\_.f64}\left(10, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 3.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left({k}^{3} \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)}, \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(k \cdot {k}^{2}\right) \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(k \cdot \left({k}^{2} \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{10}, k\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{10}, k\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(100 \cdot \frac{1}{{k}^{2}} - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(100 \cdot \frac{1}{{k}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(100 \cdot \frac{1}{{k}^{2}} + -1\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(-1 + 100 \cdot \frac{1}{{k}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \left(100 \cdot \frac{1}{{k}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \left(\frac{100 \cdot 1}{{k}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \left(\frac{100}{{k}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(100, \left({k}^{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(100, \left(k \cdot k\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

      17. *-lowering-*.f64 33.2%

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(100, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(10, k\right)\right)\right)\right) \]

   10. Simplified 33.2%

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

   if 1.24999999999999996e218 < m

   1. Initial program 63.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 2.7%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in k around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot a}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(a + -100 \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{-10}\right)\right) \cdot a\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(\left(-100 + 1\right) \cdot a\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + 10 \cdot a\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right), \color{blue}{10}\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(k \cdot -1\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot \left(-100 + 1\right)\right)\right), 10\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot -99\right)\right), 10\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 99\right), 10\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      21. metadata-eval 46.8%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 99\right), 10\right)\right)\right)\right) \]

   10. Simplified 46.8%

       \[\leadsto \color{blue}{a + k \cdot \left(a \cdot \left(k \cdot 99 + 10\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.38:\\ \;\;\;\;a \cdot \frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.19:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{1 + \frac{k \cdot \left(k \cdot \left(k \cdot \left(-1 + \frac{100}{k \cdot k}\right)\right)\right)}{10 - k}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.35:\\ \;\;\;\;a \cdot \frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 44:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.35)
   (* a (/ (+ 1.0 (/ (+ -10.0 (/ 100.0 k)) k)) (* k k)))
   (if (<= m 44.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (+ 10.0 (* k 99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.35) {
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	} else if (m <= 44.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 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 <= (-1.35d0)) then
        tmp = a * ((1.0d0 + (((-10.0d0) + (100.0d0 / k)) / k)) / (k * k))
    else if (m <= 44.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * (10.0d0 + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.35) {
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	} else if (m <= 44.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.35:
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k))
	elif m <= 44.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.35)
		tmp = Float64(a * Float64(Float64(1.0 + Float64(Float64(-10.0 + Float64(100.0 / k)) / k)) / Float64(k * k)));
	elseif (m <= 44.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.35)
		tmp = a * ((1.0 + ((-10.0 + (100.0 / k)) / k)) / (k * k));
	elseif (m <= 44.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.35], N[(a * N[(N[(1.0 + N[(N[(-10.0 + N[(100.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 44.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\color{blue}{\left(10 \cdot k\right)}, \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 10\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right)\right) \]

      2. *-lowering-*.f64 43.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 10\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right)\right) \]

   8. Simplified 43.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{1}{k \cdot \frac{10 \cdot 10 - k \cdot k}{10 - k}} \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{k \cdot \frac{100 - k \cdot k}{10 - k}} \cdot a \]

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{10 - k}{\left(100 - k \cdot k\right) \cdot k}\right), \color{blue}{a}\right) \]

   10. Applied egg-rr 43.9%

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

   11. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{100}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}}{{k}^{2}}\right)}, a\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{100}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}\right), \left({k}^{2}\right)\right), a\right) \]

   13. Simplified 75.0%

       \[\leadsto \color{blue}{\frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}} \cdot a \]

   if -1.3500000000000001 < m < 44

   1. Initial program 89.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 87.4%

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

   if 44 < m

   1. Initial program 75.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.3%

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot a}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(a + -100 \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{-10}\right)\right) \cdot a\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(\left(-100 + 1\right) \cdot a\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + 10 \cdot a\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right), \color{blue}{10}\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(k \cdot -1\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot \left(-100 + 1\right)\right)\right), 10\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot -99\right)\right), 10\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 99\right), 10\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      21. metadata-eval 28.5%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 99\right), 10\right)\right)\right)\right) \]

   10. Simplified 28.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.35:\\ \;\;\;\;a \cdot \frac{1 + \frac{-10 + \frac{100}{k}}{k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 44:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -49000:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k}}{k} \cdot -980}{k \cdot k}\\ \mathbf{elif}\;m \leq 2100:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -49000.0)
   (/ (* (/ (/ (/ a k) k) k) -980.0) (* k k))
   (if (<= m 2100.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (+ 10.0 (* k 99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -49000.0) {
		tmp = ((((a / k) / k) / k) * -980.0) / (k * k);
	} else if (m <= 2100.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 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 <= (-49000.0d0)) then
        tmp = ((((a / k) / k) / k) * (-980.0d0)) / (k * k)
    else if (m <= 2100.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * (10.0d0 + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -49000.0) {
		tmp = ((((a / k) / k) / k) * -980.0) / (k * k);
	} else if (m <= 2100.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -49000.0:
		tmp = ((((a / k) / k) / k) * -980.0) / (k * k)
	elif m <= 2100.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -49000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a / k) / k) / k) * -980.0) / Float64(k * k));
	elseif (m <= 2100.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -49000.0)
		tmp = ((((a / k) / k) / k) * -980.0) / (k * k);
	elseif (m <= 2100.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -49000.0], N[(N[(N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * -980.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2100.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -49000

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 36.9%

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

   6. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{\frac{a + -1 \cdot \frac{-1 \cdot \frac{\left(10 \cdot \frac{a}{k} + 10 \cdot \frac{a + -100 \cdot a}{k}\right) - \left(a + -100 \cdot a\right)}{k} - -10 \cdot a}{k}}{{k}^{2}}} \]

   8. Simplified 54.4%

      \[\leadsto \color{blue}{\frac{a + \frac{\frac{-990 \cdot \frac{a}{k} + a \cdot \left(\frac{10}{k} - -99\right)}{k} + a \cdot -10}{k}}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-990 \cdot a + 10 \cdot a}{{k}^{3}}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-990 \cdot a + 10 \cdot a}{\left(k \cdot k\right) \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-990 \cdot a + 10 \cdot a}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-990 + 10\right)}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot -980}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-1 \cdot 980\right)}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-1 \cdot \left(-10 + 990\right)\right)}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(a \cdot -1\right) \cdot \left(-10 + 990\right)}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot a\right) \cdot \left(-10 + 990\right)}{{k}^{2} \cdot k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       9. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot a}{{k}^{2}} \cdot \frac{-10 + 990}{k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(a\right)}{{k}^{2}} \cdot \frac{-10 + 990}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{a}{{k}^{2}}\right)\right) \cdot \frac{-10 + 990}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       12. distribute-neg-frac2 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{\mathsf{neg}\left({k}^{2}\right)} \cdot \frac{-10 + 990}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{-1 \cdot {k}^{2}} \cdot \frac{-10 + 990}{k}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       14. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-10 + 990\right)}{\left(-1 \cdot {k}^{2}\right) \cdot k}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-10 + 990\right)}{-1 \cdot \left({k}^{2} \cdot k\right)}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-10 + 990\right)}{-1 \cdot \left(\left(k \cdot k\right) \cdot k\right)}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       17. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-10 + 990\right)}{-1 \cdot {k}^{3}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       18. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot \left(-10 + 990\right)}{{k}^{3} \cdot -1}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       19. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{{k}^{3}} \cdot \frac{-10 + 990}{-1}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{{k}^{3}} \cdot \frac{980}{-1}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       21. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{{k}^{3}} \cdot -980\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       22. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{{k}^{3}} \cdot \left(-990 + 10\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   11. Simplified 64.8%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{a}{k}}{k}}{k} \cdot -980}}{k \cdot k} \]

   if -49000 < m < 2100

   1. Initial program 89.6%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 85.6%

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

   if 2100 < m

   1. Initial program 74.7%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.2%

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot a}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(a + -100 \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{-10}\right)\right) \cdot a\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(\left(-100 + 1\right) \cdot a\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + 10 \cdot a\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right), \color{blue}{10}\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(k \cdot -1\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot \left(-100 + 1\right)\right)\right), 10\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot -99\right)\right), 10\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 99\right), 10\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      21. metadata-eval 28.8%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 99\right), 10\right)\right)\right)\right) \]

   10. Simplified 28.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -49000:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k}}{k} \cdot -980}{k \cdot k}\\ \mathbf{elif}\;m \leq 2100:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2100:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.24)
   (/ a (* k k))
   (if (<= m 2100.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (+ 10.0 (* k 99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.24) {
		tmp = a / (k * k);
	} else if (m <= 2100.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 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 <= (-0.24d0)) then
        tmp = a / (k * k)
    else if (m <= 2100.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * (10.0d0 + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.24) {
		tmp = a / (k * k);
	} else if (m <= 2100.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.24:
		tmp = a / (k * k)
	elif m <= 2100.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.24)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2100.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.24)
		tmp = a / (k * k);
	elseif (m <= 2100.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.24], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2100.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.23999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 63.9%

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

   if -0.23999999999999999 < m < 2100

   1. Initial program 89.5%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 86.5%

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

   if 2100 < m

   1. Initial program 74.7%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.2%

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in k around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - -10 \cdot a\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right) \cdot a}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(a + -100 \cdot a\right) + \left(\mathsf{neg}\left(\color{blue}{-10}\right)\right) \cdot a\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(\left(-100 + 1\right) \cdot a\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + \color{blue}{\left(\mathsf{neg}\left(-10\right)\right)} \cdot a\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right) \cdot a + 10 \cdot a\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right) + 10\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(k\right)\right) \cdot \left(-100 + 1\right)\right), \color{blue}{10}\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(-1 \cdot k\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(k \cdot -1\right) \cdot \left(-100 + 1\right)\right), 10\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot \left(-100 + 1\right)\right)\right), 10\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(-1 \cdot -99\right)\right), 10\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 99\right), 10\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right)\right), 10\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, \left(\mathsf{neg}\left(-99\right)\right)\right), 10\right)\right)\right)\right) \]

      21. metadata-eval 28.8%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 99\right), 10\right)\right)\right)\right) \]

   10. Simplified 28.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2100:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 44:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.32)
   (/ a (* k k))
   (if (<= m 44.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* a (+ -10.0 (* k 99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.32) {
		tmp = a / (k * k);
	} else if (m <= 44.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 + (k * 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 <= (-0.32d0)) then
        tmp = a / (k * k)
    else if (m <= 44.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * ((-10.0d0) + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.32) {
		tmp = a / (k * k);
	} else if (m <= 44.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.32:
		tmp = a / (k * k)
	elif m <= 44.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (-10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.32)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 44.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(-10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.32)
		tmp = a / (k * k);
	elseif (m <= 44.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (-10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.32], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 44.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(-10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.320000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 63.9%

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

   if -0.320000000000000007 < m < 44

   1. Initial program 89.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 87.4%

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

   if 44 < m

   1. Initial program 75.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 3.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(10 + k\right) \cdot \color{blue}{k}\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{10 \cdot 10 - k \cdot k}{10 - k} \cdot k\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\frac{\left(10 \cdot 10 - k \cdot k\right) \cdot k}{\color{blue}{10 - k}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(10 \cdot 10 - k \cdot k\right) \cdot k\right), \color{blue}{\left(10 - k\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(10 \cdot 10 - k \cdot k\right), k\right), \left(\color{blue}{10} - k\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(10 \cdot 10\right), \left(k \cdot k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \left(k \cdot k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, k\right)\right), k\right), \left(10 - k\right)\right)\right)\right) \]

      9. --lowering--.f64 3.2%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(100, \mathsf{*.f64}\left(k, k\right)\right), k\right), \mathsf{\_.f64}\left(10, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 3.2%

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

   8. Taylor expanded in k around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{\left(\mathsf{neg}\left(10\right)\right) \cdot a}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(10\right)\right)} \cdot a\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(a + -100 \cdot a\right) + -10 \cdot a\right)\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(\left(-100 + 1\right) \cdot a\right) + -10 \cdot a\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(-1 \cdot k\right) \cdot \left(-99 \cdot a\right) + -10 \cdot a\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(\left(\left(-1 \cdot k\right) \cdot -99\right) \cdot a + \color{blue}{-10} \cdot a\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \color{blue}{\left(\left(-1 \cdot k\right) \cdot -99 + -10\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \left(\left(k \cdot -1\right) \cdot -99 + -10\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \left(k \cdot \left(-1 \cdot -99\right) + -10\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \left(a \cdot \left(k \cdot 99 + -10\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \color{blue}{\left(k \cdot 99 + -10\right)}\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(k \cdot 99\right), \color{blue}{-10}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 28.5%

          \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(k, 99\right), -10\right)\right)\right)\right) \]

   10. Simplified 28.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 44:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.46)
   (/ a (* k k))
   (if (<= m 1.32e-9)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (* a (+ 1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.46) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.46d0)) then
        tmp = a / (k * k)
    else if (m <= 1.32d-9) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (1.0d0 + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.46) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.46:
		tmp = a / (k * k)
	elif m <= 1.32e-9:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.46)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.32e-9)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.46)
		tmp = a / (k * k);
	elseif (m <= 1.32e-9)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.46], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.32e-9], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.46000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 63.9%

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

   if -0.46000000000000002 < m < 1.32e-9

   1. Initial program 89.1%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 88.8%

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

   if 1.32e-9 < m

   1. Initial program 75.6%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.8%

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

   7. Applied egg-rr 3.5%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

         \[\leadsto \left(1 + 10 \cdot k\right) \cdot a \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 10 \cdot k\right), \color{blue}{a}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(10 \cdot k\right)\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(k \cdot 10\right)\right), a\right) \]

      8. *-lowering-*.f64 13.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, 10\right)\right), a\right) \]

   10. Simplified 13.1%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.28:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.28)
   (/ a (* k k))
   (if (<= m 1.32e-9) (/ a (+ 1.0 (* k k))) (* a (+ 1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.28) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.28d0)) then
        tmp = a / (k * k)
    else if (m <= 1.32d-9) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * (1.0d0 + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.28) {
		tmp = a / (k * k);
	} else if (m <= 1.32e-9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.28:
		tmp = a / (k * k)
	elif m <= 1.32e-9:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.28)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.32e-9)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.28)
		tmp = a / (k * k);
	elseif (m <= 1.32e-9)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.28], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.32e-9], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.28000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 36.5%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 63.9%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 63.9%

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

   if -0.28000000000000003 < m < 1.32e-9

   1. Initial program 89.1%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 88.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      2. *-lowering-*.f64 88.5%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   8. Simplified 88.5%

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

   if 1.32e-9 < m

   1. Initial program 75.6%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.8%

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

   7. Applied egg-rr 3.5%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

         \[\leadsto \left(1 + 10 \cdot k\right) \cdot a \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 10 \cdot k\right), \color{blue}{a}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(10 \cdot k\right)\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(k \cdot 10\right)\right), a\right) \]

      8. *-lowering-*.f64 13.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, 10\right)\right), a\right) \]

   10. Simplified 13.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.28:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.000155:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.6e-287)
   (/ a (* k k))
   (if (<= k 0.000155) (* a (+ 1.0 (* k -10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.6e-287) {
		tmp = a / (k * k);
	} else if (k <= 0.000155) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 4.6d-287) then
        tmp = a / (k * k)
    else if (k <= 0.000155d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.6e-287) {
		tmp = a / (k * k);
	} else if (k <= 0.000155) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.6e-287:
		tmp = a / (k * k)
	elif k <= 0.000155:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.6e-287)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.000155)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.6e-287)
		tmp = a / (k * k);
	elseif (k <= 0.000155)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.6e-287], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.000155], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.59999999999999972e-287

   1. Initial program 95.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 17.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 35.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 35.0%

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

   if 4.59999999999999972e-287 < k < 1.55e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. rem-exp-log N/A

         \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot {\left(e^{\log k}\right)}^{m}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot {\left(e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log k\right)\right)\right)}\right)}^{m}\right) \]

      9. log-rec N/A

         \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot {\left(e^{\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)}\right)}^{m}\right) \]

      10. exp-prod N/A

          \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot e^{\left(\mathsf{neg}\left(\log \left(\frac{1}{k}\right)\right)\right) \cdot m}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot e^{\mathsf{neg}\left(\log \left(\frac{1}{k}\right) \cdot m\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(k \cdot -10 + 1\right) \cdot \left(a \cdot e^{\mathsf{neg}\left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right) \]

      13. mul-1-neg N/A

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

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(k \cdot -10 + 1\right), \color{blue}{\left(a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)}\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(k \cdot -10\right), 1\right), \left(\color{blue}{a} \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, -10\right), 1\right), \left(a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(k, -10\right), 1\right), \color{blue}{a}\right) \]
   7. Step-by-step derivation

   8. Simplified 54.3%

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

   if 1.55e-4 < k

   1. Initial program 72.1%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 57.0%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

      4. /-lowering-/.f64 62.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   8. Simplified 62.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.000155:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 21500:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.7e-287)
   (/ a (* k k))
   (if (<= k 21500.0) (* a (+ 1.0 (* k 10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.7e-287) {
		tmp = a / (k * k);
	} else if (k <= 21500.0) {
		tmp = a * (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 4.7d-287) then
        tmp = a / (k * k)
    else if (k <= 21500.0d0) then
        tmp = a * (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.7e-287) {
		tmp = a / (k * k);
	} else if (k <= 21500.0) {
		tmp = a * (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.7e-287:
		tmp = a / (k * k)
	elif k <= 21500.0:
		tmp = a * (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.7e-287)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 21500.0)
		tmp = Float64(a * Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.7e-287)
		tmp = a / (k * k);
	elseif (k <= 21500.0)
		tmp = a * (1.0 + (k * 10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.7e-287], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 21500.0], N[(a * N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.6999999999999999e-287

   1. Initial program 95.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 17.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 35.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 35.0%

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

   if 4.6999999999999999e-287 < k < 21500

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(10, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 52.4%

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

   7. Applied egg-rr 52.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

         \[\leadsto \left(1 + 10 \cdot k\right) \cdot a \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + 10 \cdot k\right), \color{blue}{a}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(10 \cdot k\right)\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(k \cdot 10\right)\right), a\right) \]

      8. *-lowering-*.f64 52.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, 10\right)\right), a\right) \]

   10. Simplified 52.4%

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

   if 21500 < k

   1. Initial program 71.2%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 58.8%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

      4. /-lowering-/.f64 64.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   8. Simplified 64.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 21500:\\ \;\;\;\;a \cdot \left(1 + k \cdot 10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-287}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 21500:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.25e-287) (/ a (* k k)) (if (<= k 21500.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.25e-287) {
		tmp = a / (k * k);
	} else if (k <= 21500.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.25d-287) then
        tmp = a / (k * k)
    else if (k <= 21500.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.25e-287) {
		tmp = a / (k * k);
	} else if (k <= 21500.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.25e-287:
		tmp = a / (k * k)
	elif k <= 21500.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.25e-287)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 21500.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.25e-287)
		tmp = a / (k * k);
	elseif (k <= 21500.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.25e-287], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 21500.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.25000000000000006e-287

   1. Initial program 95.4%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 17.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 35.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 35.0%

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

   if 1.25000000000000006e-287 < k < 21500

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 52.4%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   7. Step-by-step derivation

   8. Simplified 52.4%

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

   if 21500 < k

   1. Initial program 71.2%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 58.8%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a}{k}\right), \color{blue}{k}\right) \]

      4. /-lowering-/.f64 64.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(a, k\right), k\right) \]

   8. Simplified 64.4%

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

Alternative 14: 46.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-287}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 21500:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))) (if (<= k 2e-287) t_0 (if (<= k 21500.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2e-287) {
		tmp = t_0;
	} else if (k <= 21500.0) {
		tmp = 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 * k)
    if (k <= 2d-287) then
        tmp = t_0
    else if (k <= 21500.0d0) then
        tmp = 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 / (k * k);
	double tmp;
	if (k <= 2e-287) {
		tmp = t_0;
	} else if (k <= 21500.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 2e-287:
		tmp = t_0
	elif k <= 21500.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 2e-287)
		tmp = t_0;
	elseif (k <= 21500.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 2e-287)
		tmp = t_0;
	elseif (k <= 21500.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-287], t$95$0, If[LessEqual[k, 21500.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000004e-287 or 21500 < k

   1. Initial program 83.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 38.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left({k}^{2}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. *-lowering-*.f64 47.0%

         \[\leadsto \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   8. Simplified 47.0%

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

   if 2.00000000000000004e-287 < k < 21500

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

      17. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      19. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

      21. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

   5. Simplified 52.4%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a} \]
   7. Step-by-step derivation

   8. Simplified 52.4%

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

Alternative 15: 19.8% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in m around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{\left(1 + \left(10 \cdot k + {k}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \color{blue}{\left(10 + k\right)}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot 1 + k\right)\right)\right) \]

   5. lft-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + k\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + k\right)\right)\right) \]

   7. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot \color{blue}{k}\right)\right)\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(a, \left(1 + k \cdot \left(k \cdot \left(1 + \color{blue}{10 \cdot \frac{1}{k}}\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + \left(k \cdot k\right) \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \left(1 + {k}^{2} \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(k \cdot k\right) \cdot \left(\color{blue}{1} + 10 \cdot \frac{1}{k}\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \left(k \cdot \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)\right)}\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(k \cdot \left(10 \cdot \frac{1}{k} + \color{blue}{1}\right)\right)\right)\right)\right) \]

   17. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right)\right)\right)\right) \]

   18. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot \left(\frac{1}{k} \cdot k\right) + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

   19. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 \cdot 1 + 1 \cdot k\right)\right)\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + \color{blue}{1} \cdot k\right)\right)\right)\right) \]

   21. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(k, \left(10 + k\right)\right)\right)\right) \]

5. Simplified 42.9%

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

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a} \]
7. Step-by-step derivation

8. Simplified 18.9%

   \[\leadsto \color{blue}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))