Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 96.7%

Time: 8.1s

Alternatives: 12

Speedup: 1.1×

• 23.2% 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: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.9e-11) (/ a (pow k (- m))) (* (/ a k) (/ (pow k m) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.9e-11) {
		tmp = a / pow(k, -m);
	} else {
		tmp = (a / k) * (pow(k, m) / 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 <= 1.9d-11) then
        tmp = a / (k ** -m)
    else
        tmp = (a / k) * ((k ** m) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.9e-11) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = (a / k) * (Math.pow(k, m) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.9e-11:
		tmp = a / math.pow(k, -m)
	else:
		tmp = (a / k) * (math.pow(k, m) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.9e-11)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.9e-11)
		tmp = a / (k ^ -m);
	else
		tmp = (a / k) * ((k ^ m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.9e-11], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8999999999999999e-11

   1. Initial program 92.6%

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

      1. associate-/l* 92.6%

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

      2. associate-+l+ 92.6%

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

      3. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in k around 0 51.7%

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

      1. rec-exp 51.7%

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

      2. distribute-rgt-neg-in 51.7%

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

      3. exp-to-pow 99.3%

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

   6. Simplified 99.3%

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

   if 1.8999999999999999e-11 < 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-/l* 80.0%

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

      2. associate-+l+ 80.0%

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

      3. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in k around inf 79.3%

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

      1. unpow2 79.3%

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

      2. times-frac 94.9%

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

      3. rem-exp-log 92.9%

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

      4. div-exp 92.9%

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

      5. mul-1-neg 92.9%

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

      6. log-rec 92.9%

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

      7. distribute-lft-neg-in 92.9%

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

      8. remove-double-neg 92.9%

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

      9. div-exp 92.9%

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

      10. exp-to-pow 92.9%

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

      11. rem-exp-log 94.9%

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

   6. Simplified 94.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 2: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-13} \lor \neg \left(m \leq 2.6 \cdot 10^{-42}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.8e-13) (not (<= m 2.6e-42)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.8e-13) || !(m <= 2.6e-42)) {
		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 <= (-5.8d-13)) .or. (.not. (m <= 2.6d-42))) 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 <= -5.8e-13) || !(m <= 2.6e-42)) {
		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 <= -5.8e-13) or not (m <= 2.6e-42):
		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 <= -5.8e-13) || !(m <= 2.6e-42))
		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 <= -5.8e-13) || ~((m <= 2.6e-42)))
		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, -5.8e-13], N[Not[LessEqual[m, 2.6e-42]], $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 < -5.7999999999999995e-13 or 2.6e-42 < m

   1. Initial program 87.8%

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

      1. associate-/l* 87.8%

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

      2. associate-+l+ 87.8%

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

      3. *-commutative 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in k around 0 55.6%

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

      1. exp-to-pow 98.4%

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

      2. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   if -5.7999999999999995e-13 < m < 2.6e-42

   1. Initial program 88.3%

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

      1. associate-/l* 88.3%

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

      2. associate-+l+ 88.3%

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

      3. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in m around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. distribute-rgt-in 88.0%

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

      4. fma-udef 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in a around 0 88.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-13} \lor \neg \left(m \leq 2.6 \cdot 10^{-42}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 61.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -430:\\ \;\;\;\;a \cdot \frac{-1}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -430.0)
   (* a (/ -1.0 (* k (- k))))
   (if (<= m 1.9)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (+ (* k (* k 100.0)) (* k -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -430.0) {
		tmp = a * (-1.0 / (k * -k));
	} else if (m <= 1.9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-430.0d0)) then
        tmp = a * ((-1.0d0) / (k * -k))
    else if (m <= 1.9d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * ((k * (k * 100.0d0)) + (k * (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -430.0) {
		tmp = a * (-1.0 / (k * -k));
	} else if (m <= 1.9) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -430.0:
		tmp = a * (-1.0 / (k * -k))
	elif m <= 1.9:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -430.0)
		tmp = Float64(a * Float64(-1.0 / Float64(k * Float64(-k))));
	elseif (m <= 1.9)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(Float64(k * Float64(k * 100.0)) + Float64(k * -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -430.0)
		tmp = a * (-1.0 / (k * -k));
	elseif (m <= 1.9)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -430.0], N[(a * N[(-1.0 / N[(k * (-k)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(N[(k * N[(k * 100.0), $MachinePrecision]), $MachinePrecision] + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -430

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.1%

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

      1. +-commutative 40.1%

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

      2. unpow2 40.1%

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

      3. distribute-rgt-in 40.1%

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

      4. fma-udef 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in k around inf 64.9%

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

      1. unpow2 64.9%

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

   9. Simplified 64.9%

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

       1. frac-2neg 64.9%

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

       2. div-inv 66.0%

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

       3. distribute-rgt-neg-in 66.0%

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

   11. Applied egg-rr 66.0%

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

   if -430 < m < 1.8999999999999999

   1. Initial program 88.3%

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

      1. associate-/l* 88.3%

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

      2. associate-+l+ 88.3%

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

      3. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in m around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. distribute-rgt-in 84.2%

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

      4. fma-udef 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in a around 0 84.2%

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

   if 1.8999999999999999 < m

   1. Initial program 73.4%

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

      1. associate-/l* 73.4%

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

      2. associate-+l+ 73.4%

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

      3. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in k around 0 74.7%

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

      1. *-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in m around 0 3.0%

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

   8. Taylor expanded in k around 0 29.8%

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

      1. +-commutative 29.8%

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

      2. unpow2 29.8%

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

      3. associate-*r* 29.8%

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

      4. *-commutative 29.8%

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

      5. associate-*r* 29.8%

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

      6. *-commutative 29.8%

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

      7. distribute-rgt-out 31.1%

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

      8. associate-*l* 31.1%

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

   10. Simplified 31.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -430:\\ \;\;\;\;a \cdot \frac{-1}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \]

Alternative 4: 57.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -430:\\ \;\;\;\;a \cdot \frac{-1}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -430.0)
   (* a (/ -1.0 (* k (- k))))
   (if (<= m 1.04e+15) (/ a (+ 1.0 (* k (+ k 10.0)))) (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -430.0) {
		tmp = a * (-1.0 / (k * -k));
	} else if (m <= 1.04e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * -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 <= (-430.0d0)) then
        tmp = a * ((-1.0d0) / (k * -k))
    else if (m <= 1.04d+15) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -430.0) {
		tmp = a * (-1.0 / (k * -k));
	} else if (m <= 1.04e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -430.0:
		tmp = a * (-1.0 / (k * -k))
	elif m <= 1.04e+15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -430.0)
		tmp = Float64(a * Float64(-1.0 / Float64(k * Float64(-k))));
	elseif (m <= 1.04e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -430.0)
		tmp = a * (-1.0 / (k * -k));
	elseif (m <= 1.04e+15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -430.0], N[(a * N[(-1.0 / N[(k * (-k)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.04e+15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -430

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.1%

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

      1. +-commutative 40.1%

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

      2. unpow2 40.1%

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

      3. distribute-rgt-in 40.1%

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

      4. fma-udef 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in k around inf 64.9%

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

      1. unpow2 64.9%

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

   9. Simplified 64.9%

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

       1. frac-2neg 64.9%

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

       2. div-inv 66.0%

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

       3. distribute-rgt-neg-in 66.0%

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

   11. Applied egg-rr 66.0%

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

   if -430 < m < 1.04e15

   1. Initial program 86.4%

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

      1. associate-/l* 86.4%

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

      2. associate-+l+ 86.4%

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

      3. *-commutative 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in m around 0 81.3%

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

      1. +-commutative 81.3%

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

      2. unpow2 81.3%

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

      3. distribute-rgt-in 81.3%

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

      4. fma-udef 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in a around 0 81.3%

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

   if 1.04e15 < m

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. associate-+l+ 75.0%

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

      3. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 3.2%

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

      1. +-commutative 3.2%

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

      2. unpow2 3.2%

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

      3. distribute-rgt-in 3.2%

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

      4. fma-udef 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in k around 0 7.3%

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

   8. Taylor expanded in k around inf 19.9%

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

      1. *-commutative 19.9%

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

      2. associate-*r* 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -430:\\ \;\;\;\;a \cdot \frac{-1}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 5: 46.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.92 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.92e-283)
   (/ 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 <= 1.92e-283) {
		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 <= 1.92d-283) 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 <= 1.92e-283) {
		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 <= 1.92e-283:
		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 <= 1.92e-283)
		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 <= 1.92e-283)
		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, 1.92e-283], 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 < 1.92000000000000002e-283

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

      3. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in m around 0 19.5%

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

      1. +-commutative 19.5%

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

      2. unpow2 19.5%

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

      3. distribute-rgt-in 19.5%

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

      4. fma-udef 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in k around inf 35.5%

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

      1. unpow2 35.5%

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

   9. Simplified 35.5%

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

   if 1.92000000000000002e-283 < 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-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 49.5%

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

      1. +-commutative 49.5%

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

      2. unpow2 49.5%

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

      3. distribute-rgt-in 49.5%

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

      4. fma-udef 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in k around 0 49.5%

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

      1. associate-*r* 49.5%

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

      2. distribute-rgt1-in 49.5%

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

      3. *-commutative 49.5%

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

   9. Applied egg-rr 49.5%

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

   if 0.10000000000000001 < k

   1. Initial program 79.8%

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

      1. associate-/l* 79.8%

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

      2. associate-+l+ 79.8%

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

      3. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in k around inf 79.1%

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

      1. unpow2 79.1%

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

      2. times-frac 94.9%

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

      3. rem-exp-log 92.8%

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

      4. div-exp 92.8%

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

      5. mul-1-neg 92.8%

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

      6. log-rec 92.8%

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

      7. distribute-lft-neg-in 92.8%

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

      8. remove-double-neg 92.8%

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

      9. div-exp 92.8%

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

      10. exp-to-pow 92.8%

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

      11. rem-exp-log 94.9%

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

   6. Simplified 94.9%

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

      1. clear-num 93.7%

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

      2. frac-times 86.0%

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

      3. *-un-lft-identity 86.0%

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

   8. Applied egg-rr 86.0%

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

   9. Taylor expanded in m around 0 59.1%

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

       1. unpow2 59.1%

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

       2. associate-/r* 63.6%

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

   11. Simplified 63.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.92 \cdot 10^{-283}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 6: 46.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-279}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 960:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.75e-279)
   (/ a (* k k))
   (if (<= k 960.0) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.75e-279) {
		tmp = a / (k * k);
	} else if (k <= 960.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.75d-279) then
        tmp = a / (k * k)
    else if (k <= 960.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.75e-279) {
		tmp = a / (k * k);
	} else if (k <= 960.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.75e-279:
		tmp = a / (k * k)
	elif k <= 960.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.75e-279)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 960.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.75e-279)
		tmp = a / (k * k);
	elseif (k <= 960.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.75e-279], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 960.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.75000000000000005e-279

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

      3. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in m around 0 19.5%

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

      1. +-commutative 19.5%

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

      2. unpow2 19.5%

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

      3. distribute-rgt-in 19.5%

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

      4. fma-udef 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in k around inf 35.5%

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

      1. unpow2 35.5%

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

   9. Simplified 35.5%

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

   if 1.75000000000000005e-279 < k < 960

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 48.9%

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

   if 960 < k

   1. Initial program 79.6%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

      3. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in k around inf 78.9%

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

      1. unpow2 78.9%

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

      2. times-frac 94.8%

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

      3. rem-exp-log 92.7%

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

      4. div-exp 92.7%

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

      5. mul-1-neg 92.7%

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

      6. log-rec 92.7%

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

      7. distribute-lft-neg-in 92.7%

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

      8. remove-double-neg 92.7%

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

      9. div-exp 92.7%

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

      10. exp-to-pow 92.7%

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

      11. rem-exp-log 94.8%

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

   6. Simplified 94.8%

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

      1. clear-num 93.7%

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

      2. frac-times 85.8%

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

      3. *-un-lft-identity 85.8%

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

   8. Applied egg-rr 85.8%

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

   9. Taylor expanded in m around 0 59.8%

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

       1. unpow2 59.8%

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

       2. associate-/r* 64.2%

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

   11. Simplified 64.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-279}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 960:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7: 44.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-282} \lor \neg \left(k \leq 2.75 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 4.7e-282) (not (<= k 2.75e-19))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 4.7e-282) || !(k <= 2.75e-19)) {
		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 <= 4.7d-282) .or. (.not. (k <= 2.75d-19))) 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 <= 4.7e-282) || !(k <= 2.75e-19)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 4.7e-282) or not (k <= 2.75e-19):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 4.7e-282) || !(k <= 2.75e-19))
		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 <= 4.7e-282) || ~((k <= 2.75e-19)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 4.7e-282], N[Not[LessEqual[k, 2.75e-19]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 4.7e-282 or 2.7499999999999998e-19 < k

   1. Initial program 83.2%

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

      1. associate-/l* 83.2%

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

      2. associate-+l+ 83.2%

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

      3. *-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 40.2%

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

      1. +-commutative 40.2%

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

      2. unpow2 40.2%

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

      3. distribute-rgt-in 40.2%

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

      4. fma-udef 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in k around inf 47.4%

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

      1. unpow2 47.4%

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

   9. Simplified 47.4%

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

   if 4.7e-282 < k < 2.7499999999999998e-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-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 99.9%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 50.6%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-282} \lor \neg \left(k \leq 2.75 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 8: 31.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1e+18) (/ a (* k 10.0)) (if (<= m 1.04e+15) a (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1e+18) {
		tmp = a / (k * 10.0);
	} else if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = k * (a * -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 <= (-1d+18)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 1.04d+15) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1e+18) {
		tmp = a / (k * 10.0);
	} else if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1e+18:
		tmp = a / (k * 10.0)
	elif m <= 1.04e+15:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1e+18)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 1.04e+15)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1e+18)
		tmp = a / (k * 10.0);
	elseif (m <= 1.04e+15)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1e+18], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.04e+15], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1e18

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 15.6%

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

   8. Taylor expanded in k around inf 20.4%

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

      1. *-commutative 20.4%

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

   10. Simplified 20.4%

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

   if -1e18 < m < 1.04e15

   1. Initial program 86.9%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

      3. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in k around 0 51.7%

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

      1. exp-to-pow 54.1%

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

      2. *-commutative 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in m around 0 44.0%

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

   if 1.04e15 < m

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. associate-+l+ 75.0%

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

      3. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 3.2%

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

      1. +-commutative 3.2%

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

      2. unpow2 3.2%

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

      3. distribute-rgt-in 3.2%

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

      4. fma-udef 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in k around 0 7.3%

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

   8. Taylor expanded in k around inf 19.9%

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

      1. *-commutative 19.9%

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

      2. associate-*r* 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 9: 45.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-279}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{-19}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 3.4e-279) (/ a (* k k)) (if (<= k 2.75e-19) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 3.4e-279) {
		tmp = a / (k * k);
	} else if (k <= 2.75e-19) {
		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 <= 3.4d-279) then
        tmp = a / (k * k)
    else if (k <= 2.75d-19) 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 <= 3.4e-279) {
		tmp = a / (k * k);
	} else if (k <= 2.75e-19) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 3.4e-279:
		tmp = a / (k * k)
	elif k <= 2.75e-19:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 3.4e-279)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 2.75e-19)
		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 <= 3.4e-279)
		tmp = a / (k * k);
	elseif (k <= 2.75e-19)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 3.4e-279], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.75e-19], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.40000000000000015e-279

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

      3. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in m around 0 19.5%

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

      1. +-commutative 19.5%

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

      2. unpow2 19.5%

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

      3. distribute-rgt-in 19.5%

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

      4. fma-udef 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in k around inf 35.5%

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

      1. unpow2 35.5%

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

   9. Simplified 35.5%

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

   if 3.40000000000000015e-279 < k < 2.7499999999999998e-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-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 99.9%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 50.6%

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

   if 2.7499999999999998e-19 < k

   1. Initial program 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

      3. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in k around inf 79.0%

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

      1. unpow2 79.0%

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

      2. times-frac 94.1%

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

      3. rem-exp-log 92.2%

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

      4. div-exp 92.1%

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

      5. mul-1-neg 92.1%

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

      6. log-rec 92.1%

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

      7. distribute-lft-neg-in 92.1%

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

      8. remove-double-neg 92.1%

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

      9. div-exp 92.2%

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

      10. exp-to-pow 92.2%

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

      11. rem-exp-log 94.1%

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

   6. Simplified 94.1%

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

      1. clear-num 93.1%

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

      2. frac-times 85.6%

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

      3. *-un-lft-identity 85.6%

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

   8. Applied egg-rr 85.6%

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

   9. Taylor expanded in m around 0 57.9%

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

       1. unpow2 57.9%

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

       2. associate-/r* 62.2%

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

   11. Simplified 62.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-279}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{-19}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10: 25.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.04e+15) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * 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 (m <= 1.04d+15) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.04e+15:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.04e+15)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.04e+15)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.04e+15], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.04e15

   1. Initial program 93.4%

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

      1. associate-/l* 93.4%

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

      2. associate-+l+ 93.4%

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

      3. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in k around 0 54.2%

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

      1. exp-to-pow 76.5%

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

      2. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in m around 0 23.9%

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

   if 1.04e15 < m

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. associate-+l+ 75.0%

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

      3. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 3.2%

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

      1. +-commutative 3.2%

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

      2. unpow2 3.2%

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

      3. distribute-rgt-in 3.2%

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

      4. fma-udef 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in k around 0 7.3%

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

   8. Taylor expanded in k around inf 19.9%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11: 25.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.04e+15) a (* k (* a -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = k * (a * -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 <= 1.04d+15) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.04e+15) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.04e+15:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.04e+15)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.04e+15)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.04e+15], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.04e15

   1. Initial program 93.4%

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

      1. associate-/l* 93.4%

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

      2. associate-+l+ 93.4%

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

      3. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in k around 0 54.2%

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

      1. exp-to-pow 76.5%

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

      2. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in m around 0 23.9%

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

   if 1.04e15 < m

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. associate-+l+ 75.0%

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

      3. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 3.2%

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

      1. +-commutative 3.2%

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

      2. unpow2 3.2%

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

      3. distribute-rgt-in 3.2%

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

      4. fma-udef 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in k around 0 7.3%

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

   8. Taylor expanded in k around inf 19.9%

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

      1. *-commutative 19.9%

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

      2. associate-*r* 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]

Alternative 12: 19.9% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 88.0%

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

   2. associate-+l+ 88.0%

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

   3. *-commutative 88.0%

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

3. Simplified 88.0%

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

4. Taylor expanded in k around 0 53.3%

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

   1. exp-to-pow 83.5%

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

   2. *-commutative 83.5%

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

6. Simplified 83.5%

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

7. Taylor expanded in m around 0 18.0%

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

8. Final simplification 18.0%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023176 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))