Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.3%

Time: 15.1s

Alternatives: 9

Speedup: 1.0×

• 22.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+108}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+108)
     (* a (/ (pow k m) (+ 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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+108) {
		tmp = a * (pow(k, m) / (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 ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 2d+108) then
        tmp = a * ((k ** m) / (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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+108) {
		tmp = a * (Math.pow(k, m) / (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 (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+108:
		tmp = a * (math.pow(k, m) / (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 (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+108)
		tmp = Float64(a * Float64((k ^ m) / 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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+108)
		tmp = a * ((k ^ m) / (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[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+108], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 2.0000000000000001e108

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. remove-double-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. remove-double-neg 99.1%

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

      6. sqr-neg 99.1%

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

      7. associate-+l+ 99.1%

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

      8. sqr-neg 99.1%

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

      9. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   if 2.0000000000000001e108 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 64.6%

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

      1. associate-/l* 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-frac-neg2 64.6%

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

      4. distribute-neg-frac2 64.6%

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

      5. remove-double-neg 64.6%

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

      6. sqr-neg 64.6%

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

      7. associate-+l+ 64.6%

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

      8. sqr-neg 64.6%

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

      9. distribute-rgt-out 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+108}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.155 \lor \neg \left(m \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.155) (not (<= m 4e-29)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.155) || !(m <= 4e-29)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-0.155d0)) .or. (.not. (m <= 4d-29))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.155) || !(m <= 4e-29)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.155) or not (m <= 4e-29):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.155) || !(m <= 4e-29))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.155) || ~((m <= 4e-29)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.155], N[Not[LessEqual[m, 4e-29]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.154999999999999999 or 3.99999999999999977e-29 < m

   1. Initial program 90.1%

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

      1. associate-/l* 90.1%

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

      2. remove-double-neg 90.1%

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

      3. distribute-frac-neg2 90.1%

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

      4. distribute-neg-frac2 90.1%

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

      5. remove-double-neg 90.1%

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

      6. sqr-neg 90.1%

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

      7. associate-+l+ 90.1%

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

      8. sqr-neg 90.1%

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

      9. distribute-rgt-out 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in k around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -0.154999999999999999 < m < 3.99999999999999977e-29

   1. Initial program 97.7%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in m around 0 97.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.155 \lor \neg \left(m \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.22:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.42:\\ \;\;\;\;\frac{a}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.22)
     (/ a (* k k))
     (if (<= m 1.42) (/ a (+ 1.0 t_0)) (* a t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.22) {
		tmp = a / (k * k);
	} else if (m <= 1.42) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a * 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 = k * (k + 10.0d0)
    if (m <= (-0.22d0)) then
        tmp = a / (k * k)
    else if (m <= 1.42d0) then
        tmp = a / (1.0d0 + t_0)
    else
        tmp = a * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.22) {
		tmp = a / (k * k);
	} else if (m <= 1.42) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a * t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.22:
		tmp = a / (k * k)
	elif m <= 1.42:
		tmp = a / (1.0 + t_0)
	else:
		tmp = a * t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.22)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.42)
		tmp = Float64(a / Float64(1.0 + t_0));
	else
		tmp = Float64(a * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.22)
		tmp = a / (k * k);
	elseif (m <= 1.42)
		tmp = a / (1.0 + t_0);
	else
		tmp = a * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.22], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.42], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.220000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 47.3%

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

      1. add-exp-log 47.2%

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

      2. log-rec 47.2%

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

      3. +-commutative 47.2%

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

      4. log1p-define 47.2%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 47.2%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   8. Taylor expanded in k around inf 37.0%

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

      1. exp-neg 38.1%

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

      2. associate-*r/ 38.1%

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

      3. *-rgt-identity 38.1%

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

      4. *-commutative 38.1%

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

      5. exp-to-pow 62.4%

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

   10. Simplified 62.4%

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

       1. inv-pow 62.4%

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

       2. pow-pow 62.4%

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

       3. metadata-eval 62.4%

          \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

       4. pow2 62.4%

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

   12. Applied egg-rr 62.4%

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

   if -0.220000000000000001 < m < 1.4199999999999999

   1. Initial program 97.8%

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

      1. associate-/l* 97.8%

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

      2. remove-double-neg 97.8%

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

      3. distribute-frac-neg2 97.8%

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

      4. distribute-neg-frac2 97.8%

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

      5. remove-double-neg 97.8%

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

      6. sqr-neg 97.8%

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

      7. associate-+l+ 97.8%

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

      8. sqr-neg 97.8%

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

      9. distribute-rgt-out 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in m around 0 95.7%

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

   if 1.4199999999999999 < m

   1. Initial program 79.8%

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

      1. associate-/l* 79.8%

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

      2. remove-double-neg 79.8%

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

      3. distribute-frac-neg2 79.8%

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

      4. distribute-neg-frac2 79.8%

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

      5. remove-double-neg 79.8%

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

      6. sqr-neg 79.8%

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

      7. associate-+l+ 79.8%

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

      8. sqr-neg 79.8%

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

      9. distribute-rgt-out 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in m around 0 3.2%

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

      1. add-exp-log 3.2%

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

      2. log-rec 3.2%

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

      3. +-commutative 3.2%

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

      4. log1p-define 3.2%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 1.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      2. sqrt-unprod 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right) \cdot \left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}}} \]

      3. sqr-neg 35.2%

         \[\leadsto a \cdot e^{\sqrt{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      4. sqrt-unprod 34.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      5. add-sqr-sqrt 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

      6. log1p-undefine 35.2%

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

      7. rem-exp-log 35.2%

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

      8. +-commutative 35.2%

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

   9. Applied egg-rr 35.2%

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

   10. Taylor expanded in k around inf 42.2%

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

       1. *-commutative 42.2%

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

       2. unpow2 42.2%

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

       3. associate-*r* 38.9%

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

       4. distribute-lft-in 44.8%

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

       5. +-commutative 44.8%

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

       6. associate-*r* 48.1%

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

   12. Simplified 48.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.22:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.42:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k + 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k + 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.5e-299) (/ a (* k k)) (if (<= m 1.3) a (* a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.5e-299) {
		tmp = a / (k * k);
	} else if (m <= 1.3) {
		tmp = a;
	} else {
		tmp = a * (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-7.5d-299)) then
        tmp = a / (k * k)
    else if (m <= 1.3d0) then
        tmp = a
    else
        tmp = a * (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.5e-299) {
		tmp = a / (k * k);
	} else if (m <= 1.3) {
		tmp = a;
	} else {
		tmp = a * (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -7.5e-299:
		tmp = a / (k * k)
	elif m <= 1.3:
		tmp = a
	else:
		tmp = a * (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.5e-299)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.3)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -7.5e-299)
		tmp = a / (k * k);
	elseif (m <= 1.3)
		tmp = a;
	else
		tmp = a * (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.5e-299], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3], a, N[(a * N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.4999999999999999e-299

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. remove-double-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. distribute-neg-frac2 99.3%

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

      5. remove-double-neg 99.3%

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

      6. sqr-neg 99.3%

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

      7. associate-+l+ 99.3%

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

      8. sqr-neg 99.3%

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

      9. distribute-rgt-out 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in m around 0 66.2%

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

      1. add-exp-log 65.7%

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

      2. log-rec 65.7%

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

      3. +-commutative 65.7%

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

      4. log1p-define 65.7%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 65.7%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   8. Taylor expanded in k around inf 46.7%

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

      1. exp-neg 47.4%

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

      2. associate-*r/ 47.4%

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

      3. *-rgt-identity 47.4%

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

      4. *-commutative 47.4%

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

      5. exp-to-pow 62.9%

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

   10. Simplified 62.9%

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

       1. inv-pow 62.9%

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

       2. pow-pow 62.9%

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

       3. metadata-eval 62.9%

          \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

       4. pow2 62.9%

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

   12. Applied egg-rr 62.9%

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

   if -7.4999999999999999e-299 < m < 1.30000000000000004

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. remove-double-neg 97.4%

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

      3. distribute-frac-neg2 97.4%

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

      4. distribute-neg-frac2 97.4%

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

      5. remove-double-neg 97.4%

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

      6. sqr-neg 97.4%

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

      7. associate-+l+ 97.4%

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

      8. sqr-neg 97.4%

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

      9. distribute-rgt-out 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in m around 0 93.2%

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

   6. Taylor expanded in k around 0 53.1%

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

   if 1.30000000000000004 < m

   1. Initial program 79.8%

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

      1. associate-/l* 79.8%

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

      2. remove-double-neg 79.8%

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

      3. distribute-frac-neg2 79.8%

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

      4. distribute-neg-frac2 79.8%

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

      5. remove-double-neg 79.8%

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

      6. sqr-neg 79.8%

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

      7. associate-+l+ 79.8%

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

      8. sqr-neg 79.8%

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

      9. distribute-rgt-out 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in m around 0 3.2%

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

      1. add-exp-log 3.2%

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

      2. log-rec 3.2%

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

      3. +-commutative 3.2%

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

      4. log1p-define 3.2%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 1.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      2. sqrt-unprod 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right) \cdot \left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}}} \]

      3. sqr-neg 35.2%

         \[\leadsto a \cdot e^{\sqrt{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      4. sqrt-unprod 34.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      5. add-sqr-sqrt 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

      6. log1p-undefine 35.2%

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

      7. rem-exp-log 35.2%

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

      8. +-commutative 35.2%

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

   9. Applied egg-rr 35.2%

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

   10. Taylor expanded in k around inf 42.2%

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

       1. *-commutative 42.2%

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

       2. unpow2 42.2%

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

       3. associate-*r* 38.9%

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

       4. distribute-lft-in 44.8%

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

       5. +-commutative 44.8%

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

       6. associate-*r* 48.1%

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

   12. Simplified 48.1%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k + 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.45:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k + 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.1e-58)
   (/ a (* k k))
   (if (<= m 1.45) (/ a (+ 1.0 (* k 10.0))) (* a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-58) {
		tmp = a / (k * k);
	} else if (m <= 1.45) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.1d-58)) then
        tmp = a / (k * k)
    else if (m <= 1.45d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-58) {
		tmp = a / (k * k);
	} else if (m <= 1.45) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.1e-58:
		tmp = a / (k * k)
	elif m <= 1.45:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.1e-58)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.45)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.1e-58)
		tmp = a / (k * k);
	elseif (m <= 1.45)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.1e-58], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.45], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.0999999999999999e-58

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. remove-double-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. remove-double-neg 99.1%

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

      6. sqr-neg 99.1%

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

      7. associate-+l+ 99.1%

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

      8. sqr-neg 99.1%

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

      9. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in m around 0 55.0%

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

      1. add-exp-log 54.8%

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

      2. log-rec 54.8%

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

      3. +-commutative 54.8%

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

      4. log1p-define 54.8%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 54.8%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   8. Taylor expanded in k around inf 42.8%

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

      1. exp-neg 43.7%

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

      2. associate-*r/ 43.7%

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

      3. *-rgt-identity 43.7%

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

      4. *-commutative 43.7%

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

      5. exp-to-pow 64.0%

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

   10. Simplified 64.0%

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

       1. inv-pow 64.0%

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

       2. pow-pow 64.0%

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

       3. metadata-eval 64.0%

          \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

       4. pow2 64.0%

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

   12. Applied egg-rr 64.0%

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

   if -3.0999999999999999e-58 < m < 1.44999999999999996

   1. Initial program 98.6%

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

      1. associate-/l* 98.5%

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

      2. remove-double-neg 98.5%

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

      3. distribute-frac-neg2 98.5%

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

      4. distribute-neg-frac2 98.5%

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

      5. remove-double-neg 98.5%

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

      6. sqr-neg 98.5%

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

      7. associate-+l+ 98.5%

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

      8. sqr-neg 98.5%

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

      9. distribute-rgt-out 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in m around 0 96.3%

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

   6. Taylor expanded in k around 0 66.1%

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

      1. *-commutative 66.1%

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

   8. Simplified 66.1%

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

   if 1.44999999999999996 < m

   1. Initial program 79.8%

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

      1. associate-/l* 79.8%

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

      2. remove-double-neg 79.8%

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

      3. distribute-frac-neg2 79.8%

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

      4. distribute-neg-frac2 79.8%

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

      5. remove-double-neg 79.8%

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

      6. sqr-neg 79.8%

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

      7. associate-+l+ 79.8%

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

      8. sqr-neg 79.8%

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

      9. distribute-rgt-out 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in m around 0 3.2%

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

      1. add-exp-log 3.2%

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

      2. log-rec 3.2%

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

      3. +-commutative 3.2%

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

      4. log1p-define 3.2%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 1.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      2. sqrt-unprod 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right) \cdot \left(-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}}} \]

      3. sqr-neg 35.2%

         \[\leadsto a \cdot e^{\sqrt{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      4. sqrt-unprod 34.1%

         \[\leadsto a \cdot e^{\color{blue}{\sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} \cdot \sqrt{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}}} \]

      5. add-sqr-sqrt 35.2%

         \[\leadsto a \cdot e^{\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

      6. log1p-undefine 35.2%

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

      7. rem-exp-log 35.2%

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

      8. +-commutative 35.2%

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

   9. Applied egg-rr 35.2%

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

   10. Taylor expanded in k around inf 42.2%

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

       1. *-commutative 42.2%

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

       2. unpow2 42.2%

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

       3. associate-*r* 38.9%

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

       4. distribute-lft-in 44.8%

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

       5. +-commutative 44.8%

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

       6. associate-*r* 48.1%

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

   12. Simplified 48.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.45:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k + 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-296} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.3e-296) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.3e-296) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 1.3d-296) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.3e-296) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.3e-296) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.3e-296) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 1.3e-296) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.3e-296], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-296 or 1 < k

   1. Initial program 89.4%

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

      1. associate-/l* 89.4%

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

      2. remove-double-neg 89.4%

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

      3. distribute-frac-neg2 89.4%

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

      4. distribute-neg-frac2 89.4%

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

      5. remove-double-neg 89.4%

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

      6. sqr-neg 89.4%

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

      7. associate-+l+ 89.4%

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

      8. sqr-neg 89.4%

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

      9. distribute-rgt-out 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in m around 0 48.2%

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

      1. add-exp-log 47.6%

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

      2. log-rec 47.6%

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

      3. +-commutative 47.6%

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

      4. log1p-define 47.6%

         \[\leadsto a \cdot e^{-\color{blue}{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   7. Applied egg-rr 47.6%

      \[\leadsto a \cdot \color{blue}{e^{-\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)}} \]

   8. Taylor expanded in k around inf 37.7%

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

      1. exp-neg 38.2%

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

      2. associate-*r/ 38.2%

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

      3. *-rgt-identity 38.2%

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

      4. *-commutative 38.2%

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

      5. exp-to-pow 50.5%

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

   10. Simplified 50.5%

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

       1. inv-pow 50.5%

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

       2. pow-pow 50.5%

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

       3. metadata-eval 50.5%

          \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

       4. pow2 50.5%

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

   12. Applied egg-rr 50.5%

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

   if 1.3e-296 < k < 1

   1. Initial program 99.9%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 52.9%

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

   6. Taylor expanded in k around 0 49.1%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-296} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-17}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.2e-17) (* 0.1 (/ a k)) (if (<= m 6.5e+18) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.2e-17) {
		tmp = 0.1 * (a / k);
	} else if (m <= 6.5e+18) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-6.2d-17)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 6.5d+18) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.2e-17) {
		tmp = 0.1 * (a / k);
	} else if (m <= 6.5e+18) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.2e-17:
		tmp = 0.1 * (a / k)
	elif m <= 6.5e+18:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.2e-17)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 6.5e+18)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.2e-17)
		tmp = 0.1 * (a / k);
	elseif (m <= 6.5e+18)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.2e-17], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.5e+18], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.1999999999999997e-17

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 48.8%

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

   6. Taylor expanded in k around 0 16.2%

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

      1. *-commutative 16.2%

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

   8. Simplified 16.2%

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

   9. Taylor expanded in k around inf 23.5%

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

   if -6.1999999999999997e-17 < m < 6.5e18

   1. Initial program 96.7%

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

      1. associate-/l* 96.7%

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

      2. remove-double-neg 96.7%

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

      3. distribute-frac-neg2 96.7%

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

      4. distribute-neg-frac2 96.7%

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

      5. remove-double-neg 96.7%

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

      6. sqr-neg 96.7%

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

      7. associate-+l+ 96.7%

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

      8. sqr-neg 96.7%

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

      9. distribute-rgt-out 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in m around 0 92.7%

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

   6. Taylor expanded in k around 0 44.3%

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

   if 6.5e18 < m

   1. Initial program 80.2%

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

      1. associate-/l* 80.2%

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

      2. remove-double-neg 80.2%

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

      3. distribute-frac-neg2 80.2%

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

      4. distribute-neg-frac2 80.2%

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

      5. remove-double-neg 80.2%

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

      6. sqr-neg 80.2%

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

      7. associate-+l+ 80.2%

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

      8. sqr-neg 80.2%

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

      9. distribute-rgt-out 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around 0 9.5%

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

   7. Taylor expanded in k around inf 22.9%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-17}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 5.5e+17) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.5e+17) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.5d+17) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.5e+17) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.5e+17:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.5e+17)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.5e+17)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.5e+17], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5e17

   1. Initial program 98.3%

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

      1. associate-/l* 98.3%

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

      2. remove-double-neg 98.3%

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

      3. distribute-frac-neg2 98.3%

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

      4. distribute-neg-frac2 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sqr-neg 98.3%

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

      7. associate-+l+ 98.3%

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

      8. sqr-neg 98.3%

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

      9. distribute-rgt-out 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in m around 0 71.1%

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

   6. Taylor expanded in k around 0 24.6%

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

   if 5.5e17 < m

   1. Initial program 80.2%

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

      1. associate-/l* 80.2%

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

      2. remove-double-neg 80.2%

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

      3. distribute-frac-neg2 80.2%

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

      4. distribute-neg-frac2 80.2%

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

      5. remove-double-neg 80.2%

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

      6. sqr-neg 80.2%

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

      7. associate-+l+ 80.2%

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

      8. sqr-neg 80.2%

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

      9. distribute-rgt-out 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around 0 9.5%

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

   7. Taylor expanded in k around inf 22.9%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 19.3% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 92.6%

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

   1. associate-/l* 92.6%

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

   2. remove-double-neg 92.6%

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

   3. distribute-frac-neg2 92.6%

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

   4. distribute-neg-frac2 92.6%

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

   5. remove-double-neg 92.6%

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

   6. sqr-neg 92.6%

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

   7. associate-+l+ 92.6%

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

   8. sqr-neg 92.6%

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

   9. distribute-rgt-out 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in m around 0 49.6%

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

6. Taylor expanded in k around 0 18.1%

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

7. Final simplification 18.1%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))