Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 97.5%

Time: 12.9s

Alternatives: 11

Speedup: 1.1×

• 23.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 1e+150) 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 <= 1e+150) {
		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 <= 1d+150) 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 <= 1e+150) {
		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 <= 1e+150:
		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 <= 1e+150)
		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 <= 1e+150)
		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, 1e+150], t$95$1, 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))) < 9.99999999999999981e149

   1. Initial program 97.9%

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

   if 9.99999999999999981e149 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 63.2%

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

      1. associate-*r/ 63.2%

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

      2. *-commutative 63.2%

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

      3. sqr-neg 63.2%

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

      4. associate-+l+ 63.2%

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

      5. +-commutative 63.2%

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

      6. sqr-neg 63.2%

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

      7. distribute-rgt-out 63.2%

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

      8. fma-def 63.2%

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

      9. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.3%

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

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{-6} \lor \neg \left(m \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.3e-6) (not (<= m 2.9e-6)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.3e-6) || !(m <= 2.9e-6)) {
		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 <= (-3.3d-6)) .or. (.not. (m <= 2.9d-6))) 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 <= -3.3e-6) || !(m <= 2.9e-6)) {
		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 <= -3.3e-6) or not (m <= 2.9e-6):
		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 <= -3.3e-6) || !(m <= 2.9e-6))
		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 <= -3.3e-6) || ~((m <= 2.9e-6)))
		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, -3.3e-6], N[Not[LessEqual[m, 2.9e-6]], $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 < -3.30000000000000017e-6 or 2.9000000000000002e-6 < m

   1. Initial program 88.3%

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

      1. associate-*r/ 88.3%

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

      2. *-commutative 88.3%

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

      3. sqr-neg 88.3%

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

      4. associate-+l+ 88.3%

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

      5. +-commutative 88.3%

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

      6. sqr-neg 88.3%

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

      7. distribute-rgt-out 88.3%

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

      8. fma-def 88.3%

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

      9. +-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -3.30000000000000017e-6 < m < 2.9000000000000002e-6

   1. Initial program 94.4%

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

      1. associate-*r/ 94.4%

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

      2. *-commutative 94.4%

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

      3. sqr-neg 94.4%

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

      4. associate-+l+ 94.4%

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

      5. +-commutative 94.4%

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

      6. sqr-neg 94.4%

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

      7. distribute-rgt-out 94.4%

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

      8. fma-def 94.4%

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

      9. +-commutative 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in m around 0 94.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{-6} \lor \neg \left(m \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 59.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.64)
   (* a (/ 1.0 (* k k)))
   (if (<= m 64000000000000.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.64) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 64000000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} 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 <= (-0.64d0)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 64000000000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    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 <= -0.64) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 64000000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.64:
		tmp = a * (1.0 / (k * k))
	elif m <= 64000000000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.64)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 64000000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 <= -0.64)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 64000000000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.64], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 64000000000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.640000000000000013

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 54.3%

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

      4. associate-*r* 54.3%

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

      5. exp-prod 54.3%

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

      6. mul-1-neg 54.3%

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

      7. log-rec 54.3%

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

   6. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. clear-num 54.3%

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

      3. frac-times 54.3%

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

      4. *-un-lft-identity 54.3%

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

      5. add-sqr-sqrt 27.2%

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

      6. sqrt-unprod 27.4%

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

      7. sqr-neg 27.4%

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

      8. sqrt-unprod 0.2%

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

      9. add-sqr-sqrt 0.5%

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

      10. pow-exp 0.5%

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

      11. add-sqr-sqrt 0.5%

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

      12. sqrt-unprod 0.5%

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

      13. sqr-neg 0.5%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{m \cdot m}} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      14. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{\color{blue}{\left(\sqrt{m} \cdot \sqrt{m}\right)} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      15. add-sqr-sqrt 54.3%

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

      16. log-pow 100.0%

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

      17. add-exp-log 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in m around 0 53.2%

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

       1. unpow2 53.2%

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

       2. associate-/r* 39.0%

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

   11. Simplified 39.0%

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

       1. associate-/l/ 53.2%

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

       2. div-inv 54.2%

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

   13. Applied egg-rr 54.2%

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

   if -0.640000000000000013 < m < 6.4e13

   1. Initial program 94.5%

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

      1. associate-*r/ 94.5%

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

      2. *-commutative 94.5%

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

      3. sqr-neg 94.5%

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

      4. associate-+l+ 94.5%

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

      5. +-commutative 94.5%

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

      6. sqr-neg 94.5%

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

      7. distribute-rgt-out 94.5%

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

      8. fma-def 94.5%

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

      9. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in m around 0 92.1%

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

   if 6.4e13 < m

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-+l+ 75.6%

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

      5. +-commutative 75.6%

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

      6. sqr-neg 75.6%

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

      7. distribute-rgt-out 75.6%

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

      8. fma-def 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in m around 0 2.7%

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

   5. Taylor expanded in k around 0 9.3%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.64:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 4: 47.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-267}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.8e-267)
   (/ a (* k k))
   (if (<= k 0.1) (+ a (* -10.0 (* a k))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.8e-267) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (a * k));
	} 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 <= 4.8d-267) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) then
        tmp = a + ((-10.0d0) * (a * k))
    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 <= 4.8e-267) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.8e-267:
		tmp = a / (k * k)
	elif k <= 0.1:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.8e-267)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.8e-267)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.8e-267], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.7999999999999996e-267

   1. Initial program 92.9%

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

      1. associate-*r/ 92.9%

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

      2. *-commutative 92.9%

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

      3. sqr-neg 92.9%

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

      4. associate-+l+ 92.9%

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

      5. +-commutative 92.9%

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

      6. sqr-neg 92.9%

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

      7. distribute-rgt-out 92.9%

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

      8. fma-def 92.9%

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

      9. +-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in m around 0 17.7%

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

   5. Taylor expanded in k around inf 27.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 27.4%

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

   7. Simplified 27.4%

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

   if 4.7999999999999996e-267 < k < 0.10000000000000001

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.0%

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

   5. Taylor expanded in k around 0 51.4%

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

   if 0.10000000000000001 < k

   1. Initial program 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

      3. sqr-neg 79.3%

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

      4. associate-+l+ 79.3%

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

      5. +-commutative 79.3%

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

      6. sqr-neg 79.3%

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

      7. distribute-rgt-out 79.3%

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

      8. fma-def 79.3%

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

      9. +-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in k around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 90.1%

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

      4. associate-*r* 90.1%

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

      5. exp-prod 90.1%

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

      6. mul-1-neg 90.1%

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

      7. log-rec 90.1%

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

   6. Simplified 90.1%

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

      1. *-commutative 90.1%

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

      2. clear-num 89.0%

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

      3. frac-times 83.9%

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

      4. *-un-lft-identity 83.9%

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

      5. add-sqr-sqrt 38.4%

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

      6. sqrt-unprod 40.3%

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

      7. sqr-neg 40.3%

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

      8. sqrt-unprod 39.1%

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

      9. add-sqr-sqrt 39.1%

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

      10. pow-exp 39.1%

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

      11. add-sqr-sqrt 22.2%

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

      12. sqrt-unprod 54.6%

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

      13. sqr-neg 54.6%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{m \cdot m}} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      14. sqrt-unprod 32.4%

          \[\leadsto \frac{e^{\color{blue}{\left(\sqrt{m} \cdot \sqrt{m}\right)} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      15. add-sqr-sqrt 83.9%

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

      16. log-pow 83.9%

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

      17. add-exp-log 83.9%

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

   8. Applied egg-rr 83.9%

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

   9. Taylor expanded in m around 0 54.7%

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

       1. unpow2 54.7%

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

       2. associate-/r* 55.1%

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

   11. Simplified 55.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-267}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5: 47.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 67000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.4e-158)
   (* a (/ 1.0 (* k k)))
   (if (<= m 67000000000000.0) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.4e-158) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 67000000000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} 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 <= (-2.4d-158)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 67000000000000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    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 <= -2.4e-158) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 67000000000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.4e-158:
		tmp = a * (1.0 / (k * k))
	elif m <= 67000000000000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.4e-158)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 67000000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	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 <= -2.4e-158)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 67000000000000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.4e-158], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 67000000000000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.40000000000000007e-158

   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-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. sqr-neg 98.3%

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

      4. associate-+l+ 98.3%

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

      5. +-commutative 98.3%

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

      6. sqr-neg 98.3%

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

      7. distribute-rgt-out 98.3%

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

      8. fma-def 98.3%

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

      9. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in k around inf 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 56.5%

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

      4. associate-*r* 56.5%

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

      5. exp-prod 56.5%

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

      6. mul-1-neg 56.5%

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

      7. log-rec 56.5%

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

   6. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. clear-num 56.5%

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

      3. frac-times 56.4%

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

      4. *-un-lft-identity 56.4%

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

      5. add-sqr-sqrt 33.0%

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

      6. sqrt-unprod 34.1%

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

      7. sqr-neg 34.1%

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

      8. sqrt-unprod 11.6%

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

      9. add-sqr-sqrt 11.8%

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

      10. pow-exp 11.8%

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

      11. add-sqr-sqrt 11.8%

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

      12. sqrt-unprod 11.8%

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

      13. sqr-neg 11.8%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{m \cdot m}} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      14. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{\color{blue}{\left(\sqrt{m} \cdot \sqrt{m}\right)} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      15. add-sqr-sqrt 56.4%

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

      16. log-pow 94.3%

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

      17. add-exp-log 94.3%

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

   8. Applied egg-rr 94.3%

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

   9. Taylor expanded in m around 0 53.8%

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

       1. unpow2 53.8%

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

       2. associate-/r* 43.8%

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

   11. Simplified 43.8%

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

       1. associate-/l/ 53.8%

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

       2. div-inv 54.7%

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

   13. Applied egg-rr 54.7%

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

   if -2.40000000000000007e-158 < m < 6.7e13

   1. Initial program 96.0%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. sqr-neg 95.9%

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

      4. associate-+l+ 95.9%

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

      5. +-commutative 95.9%

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

      6. sqr-neg 95.9%

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

      7. distribute-rgt-out 95.9%

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

      8. fma-def 95.9%

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

      9. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in m around 0 92.7%

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

   5. Taylor expanded in k around 0 66.6%

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

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   if 6.7e13 < m

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-+l+ 75.6%

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

      5. +-commutative 75.6%

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

      6. sqr-neg 75.6%

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

      7. distribute-rgt-out 75.6%

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

      8. fma-def 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in m around 0 2.7%

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

   5. Taylor expanded in k around 0 9.3%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 67000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 6: 58.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.48:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.48)
   (* a (/ 1.0 (* k k)))
   (if (<= m 64000000000000.0) (/ a (+ 1.0 (* k k))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.48) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 64000000000000.0) {
		tmp = a / (1.0 + (k * k));
	} 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 <= (-0.48d0)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 64000000000000.0d0) then
        tmp = a / (1.0d0 + (k * k))
    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 <= -0.48) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 64000000000000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.48:
		tmp = a * (1.0 / (k * k))
	elif m <= 64000000000000.0:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.48)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 64000000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	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 <= -0.48)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 64000000000000.0)
		tmp = a / (1.0 + (k * k));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.48], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 64000000000000.0], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.47999999999999998

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 54.3%

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

      1. *-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 54.3%

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

      4. associate-*r* 54.3%

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

      5. exp-prod 54.3%

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

      6. mul-1-neg 54.3%

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

      7. log-rec 54.3%

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

   6. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. clear-num 54.3%

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

      3. frac-times 54.3%

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

      4. *-un-lft-identity 54.3%

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

      5. add-sqr-sqrt 27.2%

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

      6. sqrt-unprod 27.4%

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

      7. sqr-neg 27.4%

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

      8. sqrt-unprod 0.2%

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

      9. add-sqr-sqrt 0.5%

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

      10. pow-exp 0.5%

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

      11. add-sqr-sqrt 0.5%

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

      12. sqrt-unprod 0.5%

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

      13. sqr-neg 0.5%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{m \cdot m}} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      14. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{\color{blue}{\left(\sqrt{m} \cdot \sqrt{m}\right)} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      15. add-sqr-sqrt 54.3%

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

      16. log-pow 100.0%

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

      17. add-exp-log 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in m around 0 53.2%

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

       1. unpow2 53.2%

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

       2. associate-/r* 39.0%

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

   11. Simplified 39.0%

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

       1. associate-/l/ 53.2%

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

       2. div-inv 54.2%

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

   13. Applied egg-rr 54.2%

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

   if -0.47999999999999998 < m < 6.4e13

   1. Initial program 94.5%

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

      1. associate-*r/ 94.5%

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

      2. *-commutative 94.5%

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

      3. sqr-neg 94.5%

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

      4. associate-+l+ 94.5%

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

      5. +-commutative 94.5%

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

      6. sqr-neg 94.5%

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

      7. distribute-rgt-out 94.5%

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

      8. fma-def 94.5%

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

      9. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in m around 0 92.1%

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

   5. Taylor expanded in k around inf 88.1%

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

      1. unpow2 88.1%

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

   7. Simplified 88.1%

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

   if 6.4e13 < m

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-+l+ 75.6%

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

      5. +-commutative 75.6%

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

      6. sqr-neg 75.6%

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

      7. distribute-rgt-out 75.6%

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

      8. fma-def 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in m around 0 2.7%

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

   5. Taylor expanded in k around 0 9.3%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.48:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 7: 46.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-266} \lor \neg \left(k \leq 0.135\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 2.2e-266) (not (<= k 0.135))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.2e-266) || !(k <= 0.135)) {
		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 <= 2.2d-266) .or. (.not. (k <= 0.135d0))) 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 <= 2.2e-266) || !(k <= 0.135)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 2.2e-266) or not (k <= 0.135):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 2.2e-266) || !(k <= 0.135))
		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 <= 2.2e-266) || ~((k <= 0.135)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 2.2e-266], N[Not[LessEqual[k, 0.135]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e-266 or 0.13500000000000001 < k

   1. Initial program 86.5%

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

      1. associate-*r/ 86.5%

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

      2. *-commutative 86.5%

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

      3. sqr-neg 86.5%

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

      4. associate-+l+ 86.5%

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

      5. +-commutative 86.5%

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

      6. sqr-neg 86.5%

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

      7. distribute-rgt-out 86.5%

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

      8. fma-def 86.5%

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

      9. +-commutative 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in m around 0 35.4%

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

   5. Taylor expanded in k around inf 40.2%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 40.2%

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

   7. Simplified 40.2%

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

   if 2.2e-266 < k < 0.13500000000000001

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.0%

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

   5. Taylor expanded in k around 0 50.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-266} \lor \neg \left(k \leq 0.135\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 8: 31.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.2e-28)
   (/ a (* k 10.0))
   (if (<= m 64000000000000.0) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.2e-28) {
		tmp = a / (k * 10.0);
	} else if (m <= 64000000000000.0) {
		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 <= (-7.2d-28)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 64000000000000.0d0) 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 <= -7.2e-28) {
		tmp = a / (k * 10.0);
	} else if (m <= 64000000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -7.2e-28:
		tmp = a / (k * 10.0)
	elif m <= 64000000000000.0:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.2e-28)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 64000000000000.0)
		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 <= -7.2e-28)
		tmp = a / (k * 10.0);
	elseif (m <= 64000000000000.0)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.2e-28], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 64000000000000.0], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.1999999999999997e-28

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 33.3%

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

   5. Taylor expanded in k around 0 15.7%

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

      1. *-commutative 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in k around inf 20.1%

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

      1. *-commutative 20.1%

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

   10. Simplified 20.1%

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

   if -7.1999999999999997e-28 < m < 6.4e13

   1. Initial program 94.4%

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

      1. associate-*r/ 94.4%

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

      2. *-commutative 94.4%

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

      3. sqr-neg 94.4%

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

      4. associate-+l+ 94.4%

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

      5. +-commutative 94.4%

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

      6. sqr-neg 94.4%

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

      7. distribute-rgt-out 94.4%

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

      8. fma-def 94.4%

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

      9. +-commutative 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in m around 0 91.9%

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

   5. Taylor expanded in k around 0 52.9%

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

   if 6.4e13 < m

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-+l+ 75.6%

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

      5. +-commutative 75.6%

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

      6. sqr-neg 75.6%

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

      7. distribute-rgt-out 75.6%

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

      8. fma-def 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in m around 0 2.7%

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

   5. Taylor expanded in k around 0 9.3%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 64000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 9: 46.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-267}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.135:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5.8e-267) (/ a (* k k)) (if (<= k 0.135) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5.8e-267) {
		tmp = a / (k * k);
	} else if (k <= 0.135) {
		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 <= 5.8d-267) then
        tmp = a / (k * k)
    else if (k <= 0.135d0) 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 <= 5.8e-267) {
		tmp = a / (k * k);
	} else if (k <= 0.135) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 5.8e-267:
		tmp = a / (k * k)
	elif k <= 0.135:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5.8e-267)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.135)
		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 <= 5.8e-267)
		tmp = a / (k * k);
	elseif (k <= 0.135)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5.8e-267], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.135], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.80000000000000043e-267

   1. Initial program 92.9%

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

      1. associate-*r/ 92.9%

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

      2. *-commutative 92.9%

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

      3. sqr-neg 92.9%

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

      4. associate-+l+ 92.9%

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

      5. +-commutative 92.9%

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

      6. sqr-neg 92.9%

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

      7. distribute-rgt-out 92.9%

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

      8. fma-def 92.9%

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

      9. +-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in m around 0 17.7%

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

   5. Taylor expanded in k around inf 27.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 27.4%

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

   7. Simplified 27.4%

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

   if 5.80000000000000043e-267 < k < 0.13500000000000001

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.0%

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

   5. Taylor expanded in k around 0 50.4%

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

   if 0.13500000000000001 < k

   1. Initial program 79.3%

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

      1. associate-*r/ 79.3%

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

      2. *-commutative 79.3%

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

      3. sqr-neg 79.3%

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

      4. associate-+l+ 79.3%

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

      5. +-commutative 79.3%

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

      6. sqr-neg 79.3%

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

      7. distribute-rgt-out 79.3%

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

      8. fma-def 79.3%

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

      9. +-commutative 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in k around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 90.1%

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

      4. associate-*r* 90.1%

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

      5. exp-prod 90.1%

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

      6. mul-1-neg 90.1%

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

      7. log-rec 90.1%

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

   6. Simplified 90.1%

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

      1. *-commutative 90.1%

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

      2. clear-num 89.0%

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

      3. frac-times 83.9%

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

      4. *-un-lft-identity 83.9%

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

      5. add-sqr-sqrt 38.4%

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

      6. sqrt-unprod 40.3%

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

      7. sqr-neg 40.3%

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

      8. sqrt-unprod 39.1%

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

      9. add-sqr-sqrt 39.1%

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

      10. pow-exp 39.1%

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

      11. add-sqr-sqrt 22.2%

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

      12. sqrt-unprod 54.6%

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

      13. sqr-neg 54.6%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{m \cdot m}} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      14. sqrt-unprod 32.4%

          \[\leadsto \frac{e^{\color{blue}{\left(\sqrt{m} \cdot \sqrt{m}\right)} \cdot \log k}}{\frac{k}{a} \cdot k} \]

      15. add-sqr-sqrt 83.9%

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

      16. log-pow 83.9%

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

      17. add-exp-log 83.9%

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

   8. Applied egg-rr 83.9%

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

   9. Taylor expanded in m around 0 54.7%

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

       1. unpow2 54.7%

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

       2. associate-/r* 55.1%

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

   11. Simplified 55.1%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-267}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.135:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10: 25.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 68000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 68000000000000.0) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 68000000000000.0) {
		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 <= 68000000000000.0d0) 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 <= 68000000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 68000000000000.0:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 68000000000000.0)
		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 <= 68000000000000.0)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 68000000000000.0], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8e13

   1. Initial program 97.5%

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

      1. associate-*r/ 97.5%

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

      2. *-commutative 97.5%

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

      3. sqr-neg 97.5%

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

      4. associate-+l+ 97.5%

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

      5. +-commutative 97.5%

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

      6. sqr-neg 97.5%

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

      7. distribute-rgt-out 97.5%

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

      8. fma-def 97.5%

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

      9. +-commutative 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in m around 0 59.5%

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

   5. Taylor expanded in k around 0 25.6%

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

   if 6.8e13 < m

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-+l+ 75.6%

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

      5. +-commutative 75.6%

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

      6. sqr-neg 75.6%

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

      7. distribute-rgt-out 75.6%

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

      8. fma-def 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in m around 0 2.7%

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

   5. Taylor expanded in k around 0 9.3%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 68000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 20.2% 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 90.1%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. sqr-neg 90.1%

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

   4. associate-+l+ 90.1%

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

   5. +-commutative 90.1%

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

   6. sqr-neg 90.1%

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

   7. distribute-rgt-out 90.1%

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

   8. fma-def 90.1%

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

   9. +-commutative 90.1%

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

3. Simplified 90.1%

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

4. Taylor expanded in m around 0 40.4%

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

5. Taylor expanded in k around 0 18.2%

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

6. Final simplification 18.2%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023287 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))