Falkner and Boettcher, Appendix A

Percentage Accurate: 89.7% → 97.4%

Time: 9.1s

Alternatives: 13

Speedup: 1.1×

• 23.6% 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: 89.7% 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: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 9.6e-15)
   (* (/ a (fma k k (fma k 10.0 1.0))) (pow k m))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 9.6e-15) {
		tmp = (a / fma(k, k, fma(k, 10.0, 1.0))) * pow(k, m);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 9.6e-15)
		tmp = Float64(Float64(a / fma(k, k, fma(k, 10.0, 1.0))) * (k ^ m));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 9.6e-15], N[(N[(a / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5999999999999998e-15

   1. Initial program 96.7%

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

      1. associate-*l/ 96.7%

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

      2. +-commutative 96.7%

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

      3. fma-def 96.7%

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

      4. +-commutative 96.7%

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

      5. *-commutative 96.7%

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

      6. fma-def 96.7%

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

   3. Simplified 96.7%

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

   if 9.5999999999999998e-15 < m

   1. Initial program 69.7%

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

   2. Taylor expanded in k around 0 48.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 9.6e-15)
   (* (pow k m) (/ a (fma k (+ k 10.0) 1.0)))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 9.6e-15) {
		tmp = pow(k, m) * (a / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 9.6e-15)
		tmp = Float64((k ^ m) * Float64(a / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 9.6e-15], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5999999999999998e-15

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*r/ 96.7%

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

      3. associate-+l+ 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-rgt-out 96.7%

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

      6. fma-def 96.7%

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

      7. +-commutative 96.7%

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

   3. Simplified 96.7%

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

   if 9.5999999999999998e-15 < m

   1. Initial program 69.7%

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

   2. Taylor expanded in k around 0 48.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 5.2e-9)
   (/ a (/ (+ 1.0 (+ (* k k) (* k 10.0))) (pow k m)))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.2e-9) {
		tmp = a / ((1.0 + ((k * k) + (k * 10.0))) / pow(k, m));
	} 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 <= 5.2d-9) then
        tmp = a / ((1.0d0 + ((k * k) + (k * 10.0d0))) / (k ** m))
    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 <= 5.2e-9) {
		tmp = a / ((1.0 + ((k * k) + (k * 10.0))) / Math.pow(k, m));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.2e-9:
		tmp = a / ((1.0 + ((k * k) + (k * 10.0))) / math.pow(k, m))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.2e-9)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * k) + Float64(k * 10.0))) / (k ^ m)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.2e-9)
		tmp = a / ((1.0 + ((k * k) + (k * 10.0))) / (k ^ m));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.2e-9], N[(a / N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.2000000000000002e-9

   1. Initial program 96.7%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

      3. *-commutative 96.7%

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

   3. Simplified 96.7%

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

   if 5.2000000000000002e-9 < m

   1. Initial program 69.4%

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

   2. Taylor expanded in k around 0 48.0%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 9.6e-15) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 9.6e-15) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 9.6d-15) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 9.6e-15) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 9.6e-15:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 9.6e-15)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 9.6e-15)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 9.6e-15], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5999999999999998e-15

   1. Initial program 96.7%

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

   if 9.5999999999999998e-15 < m

   1. Initial program 69.7%

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

   2. Taylor expanded in k around 0 48.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + \mathsf{fma}\left(k, k, k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.3e-22)
   (* (pow k m) (/ (/ a k) k))
   (if (<= m 9.6e-15) (/ a (+ 1.0 (fma k k (* k 10.0)))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.3e-22) {
		tmp = pow(k, m) * ((a / k) / k);
	} else if (m <= 9.6e-15) {
		tmp = a / (1.0 + fma(k, k, (k * 10.0)));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.3e-22)
		tmp = Float64((k ^ m) * Float64(Float64(a / k) / k));
	elseif (m <= 9.6e-15)
		tmp = Float64(a / Float64(1.0 + fma(k, k, Float64(k * 10.0))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.3e-22], N[(N[Power[k, m], $MachinePrecision] * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9.6e-15], N[(a / N[(1.0 + N[(k * k + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.30000000000000037e-22

   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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. associate-/r* 100.0%

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

      2. div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -4.30000000000000037e-22 < m < 9.5999999999999998e-15

   1. Initial program 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. distribute-rgt-out 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 94.0%

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

      1. distribute-lft-in 94.0%

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

      2. fma-udef 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 9.5999999999999998e-15 < m

   1. Initial program 69.7%

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

   2. Taylor expanded in k around 0 48.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + \mathsf{fma}\left(k, k, k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 6: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.3e-22)
   (* (pow k m) (/ (/ a k) k))
   (if (<= m 6.8e-15) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.3e-22) {
		tmp = pow(k, m) * ((a / k) / k);
	} else if (m <= 6.8e-15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.3d-22)) then
        tmp = (k ** m) * ((a / k) / k)
    else if (m <= 6.8d-15) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.3e-22) {
		tmp = Math.pow(k, m) * ((a / k) / k);
	} else if (m <= 6.8e-15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.3e-22:
		tmp = math.pow(k, m) * ((a / k) / k)
	elif m <= 6.8e-15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.3e-22)
		tmp = Float64((k ^ m) * Float64(Float64(a / k) / k));
	elseif (m <= 6.8e-15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.3e-22)
		tmp = (k ^ m) * ((a / k) / k);
	elseif (m <= 6.8e-15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.3e-22], N[(N[Power[k, m], $MachinePrecision] * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.8e-15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.30000000000000037e-22

   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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. associate-/r* 100.0%

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

      2. div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -4.30000000000000037e-22 < m < 6.8000000000000001e-15

   1. Initial program 94.1%

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

      1. associate-*r/ 94.1%

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

      2. associate-+l+ 94.1%

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

      3. +-commutative 94.1%

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

      4. distribute-rgt-out 94.1%

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

      5. fma-def 94.1%

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

      6. +-commutative 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 94.0%

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

   if 6.8000000000000001e-15 < m

   1. Initial program 69.7%

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

   2. Taylor expanded in k around 0 48.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 7: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -65000 \lor \neg \left(m \leq 9.6 \cdot 10^{-15}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -65000.0) (not (<= m 9.6e-15)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -65000.0) || !(m <= 9.6e-15)) {
		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 <= (-65000.0d0)) .or. (.not. (m <= 9.6d-15))) 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 <= -65000.0) || !(m <= 9.6e-15)) {
		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 <= -65000.0) or not (m <= 9.6e-15):
		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 <= -65000.0) || !(m <= 9.6e-15))
		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 <= -65000.0) || ~((m <= 9.6e-15)))
		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, -65000.0], N[Not[LessEqual[m, 9.6e-15]], $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 < -65000 or 9.5999999999999998e-15 < m

   1. Initial program 81.7%

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

   2. Taylor expanded in k around 0 50.0%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -65000 < m < 9.5999999999999998e-15

   1. Initial program 94.3%

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

      1. associate-*r/ 94.3%

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

      2. associate-+l+ 94.3%

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

      3. +-commutative 94.3%

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

      4. distribute-rgt-out 94.3%

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

      5. fma-def 94.3%

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

      6. +-commutative 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in m around 0 94.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -65000 \lor \neg \left(m \leq 9.6 \cdot 10^{-15}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 46.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -10 \cdot \left(a \cdot k\right)\\ t_1 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -8.2 \cdot 10^{+252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.26 \cdot 10^{-200}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.1 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* -10.0 (* a k))) (t_1 (/ a (* k k))))
   (if (<= k -8.2e+252)
     t_0
     (if (<= k 2.4e-308)
       t_1
       (if (<= k 1.26e-200)
         a
         (if (<= k 8.1e-171) t_1 (if (<= k 0.1) (+ a t_0) (/ (/ a k) k))))))))

C

double code(double a, double k, double m) {
	double t_0 = -10.0 * (a * k);
	double t_1 = a / (k * k);
	double tmp;
	if (k <= -8.2e+252) {
		tmp = t_0;
	} else if (k <= 2.4e-308) {
		tmp = t_1;
	} else if (k <= 1.26e-200) {
		tmp = a;
	} else if (k <= 8.1e-171) {
		tmp = t_1;
	} else if (k <= 0.1) {
		tmp = a + t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-10.0d0) * (a * k)
    t_1 = a / (k * k)
    if (k <= (-8.2d+252)) then
        tmp = t_0
    else if (k <= 2.4d-308) then
        tmp = t_1
    else if (k <= 1.26d-200) then
        tmp = a
    else if (k <= 8.1d-171) then
        tmp = t_1
    else if (k <= 0.1d0) then
        tmp = a + t_0
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = -10.0 * (a * k);
	double t_1 = a / (k * k);
	double tmp;
	if (k <= -8.2e+252) {
		tmp = t_0;
	} else if (k <= 2.4e-308) {
		tmp = t_1;
	} else if (k <= 1.26e-200) {
		tmp = a;
	} else if (k <= 8.1e-171) {
		tmp = t_1;
	} else if (k <= 0.1) {
		tmp = a + t_0;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = -10.0 * (a * k)
	t_1 = a / (k * k)
	tmp = 0
	if k <= -8.2e+252:
		tmp = t_0
	elif k <= 2.4e-308:
		tmp = t_1
	elif k <= 1.26e-200:
		tmp = a
	elif k <= 8.1e-171:
		tmp = t_1
	elif k <= 0.1:
		tmp = a + t_0
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(-10.0 * Float64(a * k))
	t_1 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -8.2e+252)
		tmp = t_0;
	elseif (k <= 2.4e-308)
		tmp = t_1;
	elseif (k <= 1.26e-200)
		tmp = a;
	elseif (k <= 8.1e-171)
		tmp = t_1;
	elseif (k <= 0.1)
		tmp = Float64(a + t_0);
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = -10.0 * (a * k);
	t_1 = a / (k * k);
	tmp = 0.0;
	if (k <= -8.2e+252)
		tmp = t_0;
	elseif (k <= 2.4e-308)
		tmp = t_1;
	elseif (k <= 1.26e-200)
		tmp = a;
	elseif (k <= 8.1e-171)
		tmp = t_1;
	elseif (k <= 0.1)
		tmp = a + t_0;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -8.2e+252], t$95$0, If[LessEqual[k, 2.4e-308], t$95$1, If[LessEqual[k, 1.26e-200], a, If[LessEqual[k, 8.1e-171], t$95$1, If[LessEqual[k, 0.1], N[(a + t$95$0), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -8.2000000000000007e252

   1. Initial program 16.7%

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

      1. associate-*r/ 16.7%

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

      2. associate-+l+ 16.7%

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

      3. +-commutative 16.7%

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

      4. distribute-rgt-out 16.7%

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

      5. fma-def 16.7%

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

      6. +-commutative 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in m around 0 18.0%

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

   5. Taylor expanded in k around 0 41.4%

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

   6. Taylor expanded in k around inf 41.4%

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

   if -8.2000000000000007e252 < k < 2.40000000000000008e-308 or 1.26e-200 < k < 8.1e-171

   1. Initial program 98.6%

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

      1. associate-*r/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. distribute-rgt-out 98.6%

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

      5. fma-def 98.6%

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

      6. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in m around 0 17.9%

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

   5. Taylor expanded in k around inf 33.4%

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

      1. unpow2 33.4%

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

   7. Simplified 33.4%

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

   if 2.40000000000000008e-308 < k < 1.26e-200

   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 55.9%

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

   5. Taylor expanded in k around 0 55.9%

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

   if 8.1e-171 < k < 0.10000000000000001

   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 58.2%

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

   5. Taylor expanded in k around 0 58.2%

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

   if 0.10000000000000001 < k

   1. Initial program 77.4%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 59.3%

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

   5. Taylor expanded in k around inf 56.7%

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

      1. unpow2 56.7%

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

   7. Simplified 56.7%

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

      1. un-div-inv 56.7%

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

      2. associate-/l/ 58.4%

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

   9. Applied egg-rr 58.4%

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.2 \cdot 10^{+252}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.26 \cdot 10^{-200}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.1 \cdot 10^{-171}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9: 45.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.05 \cdot 10^{+254}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-310} \lor \neg \left(k \leq 5.8 \cdot 10^{-200} \lor \neg \left(k \leq 1.35 \cdot 10^{-171}\right) \land k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2.05e+254)
   (* -10.0 (* a k))
   (if (or (<= k -5e-310)
           (not (or (<= k 5.8e-200) (and (not (<= k 1.35e-171)) (<= k 1.0)))))
     (/ a (* k k))
     a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.05e+254) {
		tmp = -10.0 * (a * k);
	} else if ((k <= -5e-310) || !((k <= 5.8e-200) || (!(k <= 1.35e-171) && (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 <= (-2.05d+254)) then
        tmp = (-10.0d0) * (a * k)
    else if ((k <= (-5d-310)) .or. (.not. (k <= 5.8d-200) .or. (.not. (k <= 1.35d-171)) .and. (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 <= -2.05e+254) {
		tmp = -10.0 * (a * k);
	} else if ((k <= -5e-310) || !((k <= 5.8e-200) || (!(k <= 1.35e-171) && (k <= 1.0)))) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2.05e+254:
		tmp = -10.0 * (a * k)
	elif (k <= -5e-310) or not ((k <= 5.8e-200) or (not (k <= 1.35e-171) and (k <= 1.0))):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2.05e+254)
		tmp = Float64(-10.0 * Float64(a * k));
	elseif ((k <= -5e-310) || !((k <= 5.8e-200) || (!(k <= 1.35e-171) && (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 <= -2.05e+254)
		tmp = -10.0 * (a * k);
	elseif ((k <= -5e-310) || ~(((k <= 5.8e-200) || (~((k <= 1.35e-171)) && (k <= 1.0)))))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2.05e+254], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k, -5e-310], N[Not[Or[LessEqual[k, 5.8e-200], And[N[Not[LessEqual[k, 1.35e-171]], $MachinePrecision], LessEqual[k, 1.0]]]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]]

Derivation

1. Split input into 3 regimes

2. if k < -2.04999999999999993e254

   1. Initial program 16.7%

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

      1. associate-*r/ 16.7%

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

      2. associate-+l+ 16.7%

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

      3. +-commutative 16.7%

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

      4. distribute-rgt-out 16.7%

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

      5. fma-def 16.7%

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

      6. +-commutative 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in m around 0 18.0%

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

   5. Taylor expanded in k around 0 41.4%

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

   6. Taylor expanded in k around inf 41.4%

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

   if -2.04999999999999993e254 < k < -4.999999999999985e-310 or 5.8e-200 < k < 1.35000000000000007e-171 or 1 < k

   1. Initial program 87.2%

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

      1. associate-*r/ 87.2%

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

      2. associate-+l+ 87.2%

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

      3. +-commutative 87.2%

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

      4. distribute-rgt-out 87.2%

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

      5. fma-def 87.2%

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

      6. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in m around 0 40.2%

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

   5. Taylor expanded in k around inf 46.0%

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

      1. unpow2 46.0%

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

   7. Simplified 46.0%

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

   if -4.999999999999985e-310 < k < 5.8e-200 or 1.35000000000000007e-171 < k < 1

   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 57.2%

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

   5. Taylor expanded in k around 0 57.2%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.05 \cdot 10^{+254}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-310} \lor \neg \left(k \leq 5.8 \cdot 10^{-200} \lor \neg \left(k \leq 1.35 \cdot 10^{-171}\right) \land k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10: 46.2% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -3 \cdot 10^{+254}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-200}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -3e+254)
     (* -10.0 (* a k))
     (if (<= k -5e-310)
       t_0
       (if (<= k 1.7e-200)
         a
         (if (<= k 2e-171) t_0 (if (<= k 1.0) a (/ (/ a k) k))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -3e+254) {
		tmp = -10.0 * (a * k);
	} else if (k <= -5e-310) {
		tmp = t_0;
	} else if (k <= 1.7e-200) {
		tmp = a;
	} else if (k <= 2e-171) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= (-3d+254)) then
        tmp = (-10.0d0) * (a * k)
    else if (k <= (-5d-310)) then
        tmp = t_0
    else if (k <= 1.7d-200) then
        tmp = a
    else if (k <= 2d-171) then
        tmp = t_0
    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 t_0 = a / (k * k);
	double tmp;
	if (k <= -3e+254) {
		tmp = -10.0 * (a * k);
	} else if (k <= -5e-310) {
		tmp = t_0;
	} else if (k <= 1.7e-200) {
		tmp = a;
	} else if (k <= 2e-171) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= -3e+254:
		tmp = -10.0 * (a * k)
	elif k <= -5e-310:
		tmp = t_0
	elif k <= 1.7e-200:
		tmp = a
	elif k <= 2e-171:
		tmp = t_0
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -3e+254)
		tmp = Float64(-10.0 * Float64(a * k));
	elseif (k <= -5e-310)
		tmp = t_0;
	elseif (k <= 1.7e-200)
		tmp = a;
	elseif (k <= 2e-171)
		tmp = t_0;
	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)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= -3e+254)
		tmp = -10.0 * (a * k);
	elseif (k <= -5e-310)
		tmp = t_0;
	elseif (k <= 1.7e-200)
		tmp = a;
	elseif (k <= 2e-171)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -3e+254], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -5e-310], t$95$0, If[LessEqual[k, 1.7e-200], a, If[LessEqual[k, 2e-171], t$95$0, If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -3.00000000000000007e254

   1. Initial program 16.7%

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

      1. associate-*r/ 16.7%

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

      2. associate-+l+ 16.7%

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

      3. +-commutative 16.7%

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

      4. distribute-rgt-out 16.7%

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

      5. fma-def 16.7%

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

      6. +-commutative 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in m around 0 18.0%

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

   5. Taylor expanded in k around 0 41.4%

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

   6. Taylor expanded in k around inf 41.4%

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

   if -3.00000000000000007e254 < k < -4.999999999999985e-310 or 1.7000000000000001e-200 < k < 2e-171

   1. Initial program 98.6%

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

      1. associate-*r/ 98.6%

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

      2. associate-+l+ 98.6%

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

      3. +-commutative 98.6%

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

      4. distribute-rgt-out 98.6%

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

      5. fma-def 98.6%

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

      6. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in m around 0 17.9%

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

   5. Taylor expanded in k around inf 33.4%

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

      1. unpow2 33.4%

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

   7. Simplified 33.4%

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

   if -4.999999999999985e-310 < k < 1.7000000000000001e-200 or 2e-171 < k < 1

   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 57.2%

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

   5. Taylor expanded in k around 0 57.2%

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

   if 1 < k

   1. Initial program 77.4%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 59.3%

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

   5. Taylor expanded in k around inf 56.7%

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

      1. unpow2 56.7%

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

   7. Simplified 56.7%

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

      1. un-div-inv 56.7%

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

      2. associate-/l/ 58.4%

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

   9. Applied egg-rr 58.4%

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{+254}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-200}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-171}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11: 58.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 38:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.3e-22)
   (* a (/ 1.0 (* k k)))
   (if (<= m 38.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.3e-22) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 38.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.3d-22)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= 38.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.3e-22) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 38.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.3e-22:
		tmp = a * (1.0 / (k * k))
	elif m <= 38.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.3e-22)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 38.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.3e-22)
		tmp = a * (1.0 / (k * k));
	elseif (m <= 38.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.3e-22], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 38.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.30000000000000037e-22

   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 36.5%

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

   5. Taylor expanded in k around inf 67.8%

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

      1. unpow2 67.8%

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

   7. Simplified 67.8%

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

   if -4.30000000000000037e-22 < m < 38

   1. Initial program 94.3%

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

      1. associate-*r/ 94.3%

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

      2. associate-+l+ 94.3%

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

      3. +-commutative 94.3%

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

      4. distribute-rgt-out 94.3%

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

      5. fma-def 94.3%

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

      6. +-commutative 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in m around 0 92.7%

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

   if 38 < m

   1. Initial program 68.8%

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

      1. associate-*r/ 68.8%

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

      2. associate-+l+ 68.8%

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

      3. +-commutative 68.8%

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

      4. distribute-rgt-out 68.8%

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

      5. fma-def 68.8%

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

      6. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in m around 0 2.5%

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

   5. Taylor expanded in k around 0 9.8%

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

   6. Taylor expanded in k around inf 17.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.3 \cdot 10^{-22}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 38:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 12: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 8.5e+19) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e+19) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8.5d+19) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e+19) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.5e+19:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.5e+19)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8.5e+19)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.5e+19], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.5e19

   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/ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. +-commutative 96.2%

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

      4. distribute-rgt-out 96.2%

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

      5. fma-def 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in m around 0 67.2%

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

   5. Taylor expanded in k around 0 31.5%

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

   if 8.5e19 < m

   1. Initial program 68.5%

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

      1. associate-*r/ 68.5%

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

      2. associate-+l+ 68.5%

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

      3. +-commutative 68.5%

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

      4. distribute-rgt-out 68.5%

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

      5. fma-def 68.5%

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

      6. +-commutative 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in m around 0 2.5%

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

   5. Taylor expanded in k around 0 10.1%

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

   6. Taylor expanded in k around inf 18.1%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 13: 20.3% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 86.2%

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

   1. associate-*r/ 86.2%

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

   2. associate-+l+ 86.2%

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

   3. +-commutative 86.2%

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

   4. distribute-rgt-out 86.2%

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

   5. fma-def 86.2%

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

   6. +-commutative 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in m around 0 43.9%

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

5. Taylor expanded in k around 0 21.3%

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

6. Final simplification 21.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023229 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))