Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 97.5%

Time: 16.0s

Alternatives: 14

Speedup: 1.1×

• 23.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 2e+189) 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 <= 2e+189) {
		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 <= 2d+189) 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 <= 2e+189) {
		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 <= 2e+189:
		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 <= 2e+189)
		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 <= 2e+189)
		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, 2e+189], 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))) < 2e189

   1. Initial program 98.1%

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

   if 2e189 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 72.5%

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

      1. associate-*r/ 72.5%

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

      2. associate-+l+ 72.5%

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

      3. +-commutative 72.5%

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

      4. distribute-rgt-out 72.5%

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

      5. fma-def 72.5%

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

      6. +-commutative 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in k around 0 68.6%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+189}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.45e-11)
   (/ (pow k m) (/ (+ 1.0 (* k (+ k 10.0))) a))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.45e-11) {
		tmp = pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.45d-11) then
        tmp = (k ** m) / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.45e-11) {
		tmp = Math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.45e-11:
		tmp = math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.45e-11)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.45e-11)
		tmp = (k ^ m) / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.45e-11], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.4499999999999999e-11

   1. Initial program 97.8%

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

      1. associate-*r/ 97.8%

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

      2. associate-+l+ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-rgt-out 97.8%

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

      5. fma-def 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in a around 0 78.4%

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

      1. exp-to-pow 97.8%

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

      2. +-commutative 97.8%

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

      3. fma-udef 97.8%

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

      4. associate-/l* 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in a around 0 97.7%

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

   if 2.4499999999999999e-11 < m

   1. Initial program 82.7%

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

      1. associate-*r/ 82.7%

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

      2. associate-+l+ 82.7%

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

      3. +-commutative 82.7%

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

      4. distribute-rgt-out 82.7%

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

      5. fma-def 82.7%

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

      6. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in k around 0 53.1%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-13} \lor \neg \left(m \leq 8.7 \cdot 10^{-17}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.5e-13) (not (<= m 8.7e-17)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.5e-13) || !(m <= 8.7e-17)) {
		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 <= (-3.5d-13)) .or. (.not. (m <= 8.7d-17))) 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 <= -3.5e-13) || !(m <= 8.7e-17)) {
		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 <= -3.5e-13) or not (m <= 8.7e-17):
		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 <= -3.5e-13) || !(m <= 8.7e-17))
		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 <= -3.5e-13) || ~((m <= 8.7e-17)))
		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, -3.5e-13], N[Not[LessEqual[m, 8.7e-17]], $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 < -3.5000000000000002e-13 or 8.69999999999999962e-17 < m

   1. Initial program 91.0%

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

      1. associate-*r/ 91.0%

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

      2. associate-+l+ 91.0%

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

      3. +-commutative 91.0%

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

      4. distribute-rgt-out 91.0%

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

      5. fma-def 91.0%

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

      6. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in k around 0 53.8%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if -3.5000000000000002e-13 < m < 8.69999999999999962e-17

   1. Initial program 96.1%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 96.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-13} \lor \neg \left(m \leq 8.7 \cdot 10^{-17}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 63.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{-10 + k \cdot -100}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.19)
   (/ a (* k k))
   (if (<= m 2.4e-11)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+
      a
      (/ (* (* a k) (- 100.0 (* (* k k) 10000.0))) (+ -10.0 (* k -100.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = a / (k * k);
	} else if (m <= 2.4e-11) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (((a * k) * (100.0 - ((k * k) * 10000.0))) / (-10.0 + (k * -100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.19d0)) then
        tmp = a / (k * k)
    else if (m <= 2.4d-11) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (((a * k) * (100.0d0 - ((k * k) * 10000.0d0))) / ((-10.0d0) + (k * (-100.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = a / (k * k);
	} else if (m <= 2.4e-11) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (((a * k) * (100.0 - ((k * k) * 10000.0))) / (-10.0 + (k * -100.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.19:
		tmp = a / (k * k)
	elif m <= 2.4e-11:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (((a * k) * (100.0 - ((k * k) * 10000.0))) / (-10.0 + (k * -100.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.19)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.4e-11)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(Float64(Float64(a * k) * Float64(100.0 - Float64(Float64(k * k) * 10000.0))) / Float64(-10.0 + Float64(k * -100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.19)
		tmp = a / (k * k);
	elseif (m <= 2.4e-11)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (((a * k) * (100.0 - ((k * k) * 10000.0))) / (-10.0 + (k * -100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.19], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4e-11], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(N[(N[(a * k), $MachinePrecision] * N[(100.0 - N[(N[(k * k), $MachinePrecision] * 10000.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-10.0 + N[(k * -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.19

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 31.2%

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

   5. Taylor expanded in k around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -0.19 < m < 2.4000000000000001e-11

   1. Initial program 96.2%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 95.8%

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

   if 2.4000000000000001e-11 < m

   1. Initial program 82.7%

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

      1. associate-*r/ 82.7%

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

      2. associate-+l+ 82.7%

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

      3. +-commutative 82.7%

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

      4. distribute-rgt-out 82.7%

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

      5. fma-def 82.7%

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

      6. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in m around 0 4.0%

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

   5. Taylor expanded in k around 0 3.8%

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

   6. Taylor expanded in k around 0 27.1%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r* 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-rgt-out 30.8%

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

      6. *-commutative 30.8%

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

      7. associate-*l* 30.8%

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

      8. distribute-lft-out 30.8%

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

   8. Simplified 30.8%

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

      1. associate-*r* 26.2%

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

      2. flip-+ 30.8%

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

      3. associate-*r/ 31.9%

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

      4. metadata-eval 31.9%

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

      5. swap-sqr 31.9%

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

      6. metadata-eval 31.9%

         \[\leadsto a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot \color{blue}{10000}\right)}{-10 - k \cdot 100} \]

      7. *-commutative 31.9%

         \[\leadsto a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{-10 - \color{blue}{100 \cdot k}} \]

      8. cancel-sign-sub-inv 31.9%

         \[\leadsto a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{\color{blue}{-10 + \left(-100\right) \cdot k}} \]

      9. metadata-eval 31.9%

         \[\leadsto a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{-10 + \color{blue}{-100} \cdot k} \]

   10. Applied egg-rr 31.9%

       \[\leadsto a + \color{blue}{\frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{-10 + -100 \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + \frac{\left(a \cdot k\right) \cdot \left(100 - \left(k \cdot k\right) \cdot 10000\right)}{-10 + k \cdot -100}\\ \end{array} \]

Alternative 5: 60.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.182:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + 100 \cdot \left(k \cdot \left(a \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.182)
   (/ a (* k k))
   (if (<= m 2.45e-11) (/ a (+ 1.0 (* k k))) (+ a (* 100.0 (* k (* a k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.182) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (100.0 * (k * (a * k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.182d0)) then
        tmp = a / (k * k)
    else if (m <= 2.45d-11) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + (100.0d0 * (k * (a * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.182) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (100.0 * (k * (a * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.182:
		tmp = a / (k * k)
	elif m <= 2.45e-11:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (100.0 * (k * (a * k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.182)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.45e-11)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(100.0 * Float64(k * Float64(a * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.182)
		tmp = a / (k * k);
	elseif (m <= 2.45e-11)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (100.0 * (k * (a * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.182], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.45e-11], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(100.0 * N[(k * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.182

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 31.2%

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

   5. Taylor expanded in k around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -0.182 < m < 2.4499999999999999e-11

   1. Initial program 96.2%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 95.8%

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

   5. Taylor expanded in k around inf 92.8%

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

      1. unpow2 92.8%

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

   7. Simplified 92.8%

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

   if 2.4499999999999999e-11 < m

   1. Initial program 82.7%

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

      1. associate-*r/ 82.7%

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

      2. associate-+l+ 82.7%

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

      3. +-commutative 82.7%

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

      4. distribute-rgt-out 82.7%

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

      5. fma-def 82.7%

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

      6. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in m around 0 4.0%

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

   5. Taylor expanded in k around 0 3.8%

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

   6. Taylor expanded in k around 0 27.1%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r* 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-rgt-out 30.8%

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

      6. *-commutative 30.8%

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

      7. associate-*l* 30.8%

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

      8. distribute-lft-out 30.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in k around inf 30.8%

      \[\leadsto a + \color{blue}{100 \cdot \left({k}^{2} \cdot a\right)} \]
   10. Step-by-step derivation

       1. unpow2 30.8%

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

       2. associate-*l* 26.2%

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

       3. *-commutative 26.2%

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

   11. Simplified 26.2%

       \[\leadsto a + \color{blue}{100 \cdot \left(k \cdot \left(a \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.182:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + 100 \cdot \left(k \cdot \left(a \cdot k\right)\right)\\ \end{array} \]

Alternative 6: 61.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.175:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.175)
   (/ a (* k k))
   (if (<= m 2.45e-11) (/ a (+ 1.0 (* k k))) (+ a (* a (* k (* k 100.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.175) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (a * (k * (k * 100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.175d0)) then
        tmp = a / (k * k)
    else if (m <= 2.45d-11) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + (a * (k * (k * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.175) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (a * (k * (k * 100.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.175:
		tmp = a / (k * k)
	elif m <= 2.45e-11:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (a * (k * (k * 100.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.175)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.45e-11)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(k * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.175)
		tmp = a / (k * k);
	elseif (m <= 2.45e-11)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (a * (k * (k * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.175], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.45e-11], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(k * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.17499999999999999

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 31.2%

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

   5. Taylor expanded in k around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -0.17499999999999999 < m < 2.4499999999999999e-11

   1. Initial program 96.2%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 95.8%

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

   5. Taylor expanded in k around inf 92.8%

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

      1. unpow2 92.8%

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

   7. Simplified 92.8%

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

   if 2.4499999999999999e-11 < m

   1. Initial program 82.7%

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

      1. associate-*r/ 82.7%

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

      2. associate-+l+ 82.7%

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

      3. +-commutative 82.7%

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

      4. distribute-rgt-out 82.7%

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

      5. fma-def 82.7%

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

      6. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in m around 0 4.0%

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

   5. Taylor expanded in k around 0 3.8%

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

   6. Taylor expanded in k around 0 27.1%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r* 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-rgt-out 30.8%

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

      6. *-commutative 30.8%

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

      7. associate-*l* 30.8%

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

      8. distribute-lft-out 30.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in k around inf 30.8%

      \[\leadsto a + \color{blue}{100 \cdot \left({k}^{2} \cdot a\right)} \]
   10. Step-by-step derivation

       1. unpow2 30.8%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-*r* 30.8%

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

       5. *-commutative 30.8%

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

   11. Simplified 30.8%

       \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(k \cdot 100\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.175:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \]

Alternative 7: 62.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.25)
   (/ a (* k k))
   (if (<= m 2.45e-11)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (* k (* k 100.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.25) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * (k * 100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.25d0)) then
        tmp = a / (k * k)
    else if (m <= 2.45d-11) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * (k * (k * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.25) {
		tmp = a / (k * k);
	} else if (m <= 2.45e-11) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * (k * 100.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.25:
		tmp = a / (k * k)
	elif m <= 2.45e-11:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * (k * (k * 100.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.25)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.45e-11)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(k * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.25)
		tmp = a / (k * k);
	elseif (m <= 2.45e-11)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * (k * (k * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.25], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.45e-11], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(k * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.25

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 31.2%

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

   5. Taylor expanded in k around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -0.25 < m < 2.4499999999999999e-11

   1. Initial program 96.2%

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

      1. associate-*r/ 96.1%

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

      2. associate-+l+ 96.1%

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

      3. +-commutative 96.1%

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

      4. distribute-rgt-out 96.1%

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

      5. fma-def 96.1%

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

      6. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 95.8%

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

   if 2.4499999999999999e-11 < m

   1. Initial program 82.7%

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

      1. associate-*r/ 82.7%

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

      2. associate-+l+ 82.7%

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

      3. +-commutative 82.7%

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

      4. distribute-rgt-out 82.7%

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

      5. fma-def 82.7%

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

      6. +-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in m around 0 4.0%

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

   5. Taylor expanded in k around 0 3.8%

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

   6. Taylor expanded in k around 0 27.1%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r* 27.1%

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

      4. *-commutative 27.1%

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

      5. distribute-rgt-out 30.8%

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

      6. *-commutative 30.8%

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

      7. associate-*l* 30.8%

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

      8. distribute-lft-out 30.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in k around inf 30.8%

      \[\leadsto a + \color{blue}{100 \cdot \left({k}^{2} \cdot a\right)} \]
   10. Step-by-step derivation

       1. unpow2 30.8%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-*r* 30.8%

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

       5. *-commutative 30.8%

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

   11. Simplified 30.8%

       \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(k \cdot 100\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \]

Alternative 8: 47.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.7e-284)
   (/ a (* k k))
   (if (<= k 0.1) (* a (+ 1.0 (* k -10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.7d-284)) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.7e-284:
		tmp = a / (k * k)
	elif k <= 0.1:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.7e-284)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.7e-284)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.7e-284], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.69999999999999996e-284

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. distribute-rgt-out 95.7%

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

      5. fma-def 95.7%

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

      6. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 20.4%

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

   5. Taylor expanded in k around inf 28.1%

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

      1. unpow2 28.1%

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

   7. Simplified 28.1%

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

   if -1.69999999999999996e-284 < 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/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 52.8%

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

   5. Taylor expanded in k around 0 51.9%

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

      1. *-commutative 51.9%

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

   7. Simplified 51.9%

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

   if 0.10000000000000001 < k

   1. Initial program 80.0%

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

      1. associate-*r/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-rgt-out 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in m around 0 67.3%

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

   5. Taylor expanded in k around inf 66.2%

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

      1. unpow2 66.2%

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

   7. Simplified 66.2%

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

      1. associate-/r* 66.3%

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

      2. associate-*r/ 70.1%

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

      3. div-inv 70.1%

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

   9. Applied egg-rr 70.1%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9: 47.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.7e-284)
   (/ a (* k k))
   (if (<= k 10.2) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 10.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.7d-284)) then
        tmp = a / (k * k)
    else if (k <= 10.2d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 10.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.7e-284:
		tmp = a / (k * k)
	elif k <= 10.2:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.7e-284)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 10.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.7e-284)
		tmp = a / (k * k);
	elseif (k <= 10.2)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.7e-284], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.2], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.69999999999999996e-284

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. distribute-rgt-out 95.7%

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

      5. fma-def 95.7%

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

      6. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 20.4%

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

   5. Taylor expanded in k around inf 28.1%

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

      1. unpow2 28.1%

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

   7. Simplified 28.1%

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

   if -1.69999999999999996e-284 < k < 10.199999999999999

   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/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 52.8%

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

   5. Taylor expanded in k around 0 52.1%

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

   if 10.199999999999999 < k

   1. Initial program 80.0%

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

      1. associate-*r/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-rgt-out 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in m around 0 67.3%

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

   5. Taylor expanded in k around inf 66.2%

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

      1. unpow2 66.2%

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

   7. Simplified 66.2%

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

      1. associate-/r* 66.3%

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

      2. associate-*r/ 70.1%

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

      3. div-inv 70.1%

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

   9. Applied egg-rr 70.1%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10: 46.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -1.7e-284) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.7e-284) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-1.7d-284)) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.7e-284) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -1.7e-284) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -1.7e-284) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -1.7e-284) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -1.7e-284], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -1.69999999999999996e-284 or 1 < k

   1. Initial program 87.6%

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

      1. associate-*r/ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. fma-def 87.6%

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

      6. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in m around 0 44.5%

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

   5. Taylor expanded in k around inf 47.7%

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

      1. unpow2 47.7%

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

   7. Simplified 47.7%

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

   if -1.69999999999999996e-284 < k < 1

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 52.8%

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

   5. Taylor expanded in k around 0 50.7%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 47.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.7e-284) (/ a (* k k)) (if (<= k 1.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.7d-284)) then
        tmp = a / (k * k)
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.7e-284) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.7e-284:
		tmp = a / (k * k)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.7e-284)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.7e-284)
		tmp = a / (k * k);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.7e-284], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.69999999999999996e-284

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. distribute-rgt-out 95.7%

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

      5. fma-def 95.7%

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

      6. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 20.4%

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

   5. Taylor expanded in k around inf 28.1%

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

      1. unpow2 28.1%

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

   7. Simplified 28.1%

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

   if -1.69999999999999996e-284 < k < 1

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 52.8%

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

   5. Taylor expanded in k around 0 50.7%

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

   if 1 < k

   1. Initial program 80.0%

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

      1. associate-*r/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-rgt-out 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in m around 0 67.3%

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

   5. Taylor expanded in k around inf 66.2%

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

      1. unpow2 66.2%

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

   7. Simplified 66.2%

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

      1. associate-/r* 66.3%

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

      2. associate-*r/ 70.1%

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

      3. div-inv 70.1%

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

   9. Applied egg-rr 70.1%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12: 51.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.076:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.076) (/ a (* k k)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.076) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (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 (m <= (-0.076d0)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.076) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.076:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.076)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.076)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.076], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0759999999999999981

   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. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 31.2%

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

   5. Taylor expanded in k around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -0.0759999999999999981 < m

   1. Initial program 90.2%

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

      1. associate-*r/ 90.2%

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

      2. associate-+l+ 90.2%

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

      3. +-commutative 90.2%

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

      4. distribute-rgt-out 90.2%

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

      5. fma-def 90.2%

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

      6. +-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in m around 0 55.0%

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

   5. Taylor expanded in k around inf 53.3%

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

      1. unpow2 53.3%

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

   7. Simplified 53.3%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.076:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 13: 26.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 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. associate-+l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. distribute-rgt-out 98.3%

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

      5. fma-def 98.3%

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

      6. +-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in m around 0 40.3%

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

   5. Taylor expanded in k around 0 32.8%

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

   if 0.10000000000000001 < k

   1. Initial program 80.0%

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

      1. associate-*r/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. +-commutative 79.9%

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

      4. distribute-rgt-out 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in m around 0 67.3%

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

   5. Taylor expanded in k around 0 26.9%

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

   6. Taylor expanded in k around inf 26.9%

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

4. Final simplification 31.1%

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

Alternative 14: 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 93.0%

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

   1. associate-*r/ 93.0%

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

   2. associate-+l+ 93.0%

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

   3. +-commutative 93.0%

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

   4. distribute-rgt-out 93.0%

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

   5. fma-def 93.0%

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

   6. +-commutative 93.0%

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

3. Simplified 93.0%

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

4. Taylor expanded in m around 0 48.1%

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

5. Taylor expanded in k around 0 24.6%

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

6. Final simplification 24.6%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023187 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))