Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.0%

Time: 7.6s

Alternatives: 13

Speedup: 1.1×

• 23.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 5e+66) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+66) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 5d+66) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+66) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 5e+66:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e+66)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 5e+66)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+66], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 4.99999999999999991e66

   1. Initial program 97.0%

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

   if 4.99999999999999991e66 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 62.7%

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

   2. Taylor expanded in k around 0 74.6%

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]
   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 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{-10} \lor \neg \left(m \leq 0.00041\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.7e-10) (not (<= m 0.00041)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.7e-10) || !(m <= 0.00041)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3.7d-10)) .or. (.not. (m <= 0.00041d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.7e-10) || !(m <= 0.00041)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -3.7e-10) or not (m <= 0.00041):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3.7e-10) || !(m <= 0.00041))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -3.7e-10) || ~((m <= 0.00041)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3.7e-10], N[Not[LessEqual[m, 0.00041]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.70000000000000015e-10 or 4.0999999999999999e-4 < m

   1. Initial program 84.9%

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

   2. Taylor expanded in k around 0 60.2%

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]
   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 -3.70000000000000015e-10 < m < 4.0999999999999999e-4

   1. Initial program 93.6%

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

      1. add-cube-cbrt 93.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      2. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      3. pow2 93.4%

         \[\leadsto \frac{a}{\color{blue}{{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right)}^{2}}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. associate-+l+ 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      5. +-commutative 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. distribute-rgt-out 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. fma-def 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. associate-+l+ 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}} \]

      9. +-commutative 93.4%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}} \]

      10. distribute-rgt-out 93.4%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}} \]

      11. fma-def 93.4%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   3. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   4. Taylor expanded in m around 0 93.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{-10} \lor \neg \left(m \leq 0.00041\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 62.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 + -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+14)
   (/ a (* k k))
   (if (<= m 1.95)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (* k (+ (* k 100.0) -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 100.0) + -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 <= (-5.2d+14)) then
        tmp = a / (k * k)
    else if (m <= 1.95d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * (k * ((k * 100.0d0) + (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 100.0) + -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+14:
		tmp = a / (k * k)
	elif m <= 1.95:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * (k * ((k * 100.0) + -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.95)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(Float64(k * 100.0) + -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+14)
		tmp = a / (k * k);
	elseif (m <= 1.95)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * (k * ((k * 100.0) + -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.95], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(N[(k * 100.0), $MachinePrecision] + -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.6%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

   if -5.2e14 < m < 1.94999999999999996

   1. Initial program 93.8%

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

      1. add-cube-cbrt 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      3. pow2 93.6%

         \[\leadsto \frac{a}{\color{blue}{{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right)}^{2}}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      5. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. distribute-rgt-out 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. fma-def 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}} \]

      9. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}} \]

      10. distribute-rgt-out 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}} \]

      11. fma-def 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   3. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   4. Taylor expanded in m around 0 91.7%

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

   if 1.94999999999999996 < m

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 85.9%

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

   3. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. fma-udef 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in k around 0 26.9%

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

      1. +-commutative 26.9%

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

      2. unpow2 26.9%

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

      3. associate-*r* 26.9%

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

      4. associate-*r* 26.9%

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

      5. *-commutative 26.9%

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

      6. distribute-rgt-out 38.5%

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

   8. Simplified 38.5%

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

      1. associate-*r* 38.5%

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

      2. *-commutative 38.5%

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

      3. distribute-rgt-out 38.5%

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

      4. *-commutative 38.5%

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

   10. Applied egg-rr 38.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 + -10\right)\right)\\ \end{array} \]

Alternative 4: 62.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 - -10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+14)
   (/ a (* k k))
   (if (<= m 2.2)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (* k (- (* k 100.0) -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 2.2) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 100.0) - -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 <= (-5.2d+14)) then
        tmp = a / (k * k)
    else if (m <= 2.2d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * (k * ((k * 100.0d0) - (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 2.2) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * (k * ((k * 100.0) - -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+14:
		tmp = a / (k * k)
	elif m <= 2.2:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * (k * ((k * 100.0) - -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(Float64(k * 100.0) - -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+14)
		tmp = a / (k * k);
	elseif (m <= 2.2)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * (k * ((k * 100.0) - -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(N[(k * 100.0), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.6%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

   if -5.2e14 < m < 2.2000000000000002

   1. Initial program 93.8%

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

      1. add-cube-cbrt 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      3. pow2 93.6%

         \[\leadsto \frac{a}{\color{blue}{{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right)}^{2}}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      5. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. distribute-rgt-out 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. fma-def 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}} \]

      9. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}} \]

      10. distribute-rgt-out 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}} \]

      11. fma-def 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   3. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   4. Taylor expanded in m around 0 91.7%

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

   if 2.2000000000000002 < m

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 85.9%

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

   3. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. fma-udef 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in k around 0 26.9%

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

      1. +-commutative 26.9%

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

      2. unpow2 26.9%

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

      3. associate-*r* 26.9%

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

      4. associate-*r* 26.9%

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

      5. *-commutative 26.9%

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

      6. distribute-rgt-out 38.5%

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

   8. Simplified 38.5%

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

      1. add-sqr-sqrt 6.5%

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

      2. sqrt-unprod 38.5%

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

      3. swap-sqr 38.5%

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

      4. metadata-eval 38.5%

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

      5. sqrt-prod 38.5%

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

      6. sqrt-prod 31.9%

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

      7. add-sqr-sqrt 38.5%

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

      8. metadata-eval 38.5%

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

      9. *-commutative 38.5%

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

      10. metadata-eval 38.5%

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

      11. cancel-sign-sub-inv 38.5%

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

      12. associate-*r* 38.5%

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

      13. distribute-rgt-out-- 38.5%

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

      14. *-commutative 38.5%

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

   10. Applied egg-rr 38.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 - -10\right)\right)\\ \end{array} \]

Alternative 5: 59.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+14)
   (/ a (* k k))
   (if (<= m 1.9) (/ a (+ 1.0 (* k k))) (+ a (* k (* k (* a 100.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 1.9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (k * (k * (a * 100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5.2d+14)) then
        tmp = a / (k * k)
    else if (m <= 1.9d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + (k * (k * (a * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 1.9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (k * (k * (a * 100.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+14:
		tmp = a / (k * k)
	elif m <= 1.9:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (k * (k * (a * 100.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.9)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(k * Float64(k * Float64(a * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+14)
		tmp = a / (k * k);
	elseif (m <= 1.9)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (k * (k * (a * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(k * N[(a * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.6%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

   if -5.2e14 < m < 1.8999999999999999

   1. Initial program 93.8%

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

   2. Taylor expanded in m around 0 91.7%

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

   3. Taylor expanded in k around 0 90.0%

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

   if 1.8999999999999999 < m

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 85.9%

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

   3. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. fma-udef 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in k around 0 26.9%

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

      1. +-commutative 26.9%

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

      2. unpow2 26.9%

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

      3. associate-*r* 26.9%

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

      4. associate-*r* 26.9%

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

      5. *-commutative 26.9%

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

      6. distribute-rgt-out 38.5%

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

   8. Simplified 38.5%

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

   9. Taylor expanded in k around inf 38.5%

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

       1. unpow2 38.5%

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

       2. associate-*r* 38.5%

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

       3. *-commutative 38.5%

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

       4. associate-*r* 38.5%

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

       5. associate-*l* 30.1%

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

       6. *-commutative 30.1%

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

   11. Simplified 30.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 100\right)\right)\\ \end{array} \]

Alternative 6: 60.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+14)
   (/ a (* k k))
   (if (<= m 2.2)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* k (* k (* a 100.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 2.2) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (k * (a * 100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5.2d+14)) then
        tmp = a / (k * k)
    else if (m <= 2.2d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (k * (a * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else if (m <= 2.2) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (k * (a * 100.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+14:
		tmp = a / (k * k)
	elif m <= 2.2:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (k * (a * 100.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+14)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(k * Float64(a * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+14)
		tmp = a / (k * k);
	elseif (m <= 2.2)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (k * (a * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(k * N[(a * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.6%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

   if -5.2e14 < m < 2.2000000000000002

   1. Initial program 93.8%

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

      1. add-cube-cbrt 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{a}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      3. pow2 93.6%

         \[\leadsto \frac{a}{\color{blue}{{\left(\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}\right)}^{2}}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      5. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. distribute-rgt-out 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. fma-def 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. associate-+l+ 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}} \]

      9. +-commutative 93.6%

         \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}} \]

      10. distribute-rgt-out 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{k \cdot \left(10 + k\right)} + 1}} \]

      11. fma-def 93.6%

          \[\leadsto \frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   3. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{a}{{\left(\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)}^{2}} \cdot \frac{{k}^{m}}{\sqrt[3]{\mathsf{fma}\left(k, 10 + k, 1\right)}}} \]

   4. Taylor expanded in m around 0 91.7%

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

   if 2.2000000000000002 < m

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 85.9%

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

   3. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. fma-udef 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in k around 0 26.9%

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

      1. +-commutative 26.9%

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

      2. unpow2 26.9%

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

      3. associate-*r* 26.9%

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

      4. associate-*r* 26.9%

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

      5. *-commutative 26.9%

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

      6. distribute-rgt-out 38.5%

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

   8. Simplified 38.5%

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

   9. Taylor expanded in k around inf 38.5%

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

       1. unpow2 38.5%

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

       2. associate-*r* 38.5%

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

       3. *-commutative 38.5%

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

       4. associate-*r* 38.5%

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

       5. associate-*l* 30.1%

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

       6. *-commutative 30.1%

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

   11. Simplified 30.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(k \cdot \left(a \cdot 100\right)\right)\\ \end{array} \]

Alternative 7: 46.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 9.2e-300)
   (/ a (* k k))
   (if (<= k 5e-9) (* a (+ 1.0 (* k -10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 9.2e-300) {
		tmp = a / (k * k);
	} else if (k <= 5e-9) {
		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 <= 9.2d-300) then
        tmp = a / (k * k)
    else if (k <= 5d-9) 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 <= 9.2e-300) {
		tmp = a / (k * k);
	} else if (k <= 5e-9) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 9.2e-300:
		tmp = a / (k * k)
	elif k <= 5e-9:
		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 <= 9.2e-300)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 5e-9)
		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 <= 9.2e-300)
		tmp = a / (k * k);
	elseif (k <= 5e-9)
		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, 9.2e-300], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e-9], 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 < 9.20000000000000003e-300

   1. Initial program 95.5%

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

   2. Taylor expanded in m around 0 23.6%

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

   3. Taylor expanded in k around inf 35.9%

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

      1. unpow2 35.9%

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

   5. Simplified 35.9%

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

   if 9.20000000000000003e-300 < k < 5.0000000000000001e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in k around 0 72.1%

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

      1. exp-to-pow 72.1%

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

      2. exp-to-pow 72.1%

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

      3. exp-to-pow 72.1%

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

      4. *-commutative 72.1%

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

      5. associate-*r* 72.1%

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

      6. exp-to-pow 72.1%

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

      7. *-commutative 72.1%

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

      8. distribute-lft1-in 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around 0 53.7%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 73.1%

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

   2. Taylor expanded in m around 0 54.2%

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

   3. Taylor expanded in k around inf 52.9%

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

      1. unpow2 52.9%

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

   5. Simplified 52.9%

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

   6. Taylor expanded in a around 0 52.9%

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

      1. unpow2 52.9%

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

      2. associate-/r* 57.5%

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

   8. Simplified 57.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 9.8e-300)
   (/ a (* k k))
   (if (<= k 10.0) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 9.8e-300) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		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 <= 9.8d-300) then
        tmp = a / (k * k)
    else if (k <= 10.0d0) 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 <= 9.8e-300) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 9.8e-300:
		tmp = a / (k * k)
	elif k <= 10.0:
		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 <= 9.8e-300)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 10.0)
		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 <= 9.8e-300)
		tmp = a / (k * k);
	elseif (k <= 10.0)
		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, 9.8e-300], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.0], 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 < 9.8e-300

   1. Initial program 95.5%

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

   2. Taylor expanded in m around 0 23.6%

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

   3. Taylor expanded in k around inf 35.9%

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

      1. unpow2 35.9%

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

   5. Simplified 35.9%

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

   if 9.8e-300 < k < 10

   1. Initial program 99.9%

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

   2. Taylor expanded in k around 0 99.1%

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

   3. Taylor expanded in m around 0 52.7%

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

   if 10 < k

   1. Initial program 72.6%

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

   2. Taylor expanded in m around 0 54.3%

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

   3. Taylor expanded in k around inf 53.8%

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

      1. unpow2 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in a around 0 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/r* 58.5%

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

   8. Simplified 58.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9: 27.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{+94} \lor \neg \left(k \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -9e+94) (not (<= k 5e-9))) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9e+94) || !(k <= 5e-9)) {
		tmp = (a / k) * 0.1;
	} 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 <= (-9d+94)) .or. (.not. (k <= 5d-9))) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9e+94) || !(k <= 5e-9)) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -9e+94) or not (k <= 5e-9):
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -9e+94) || !(k <= 5e-9))
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -9e+94) || ~((k <= 5e-9)))
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -9e+94], N[Not[LessEqual[k, 5e-9]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -8.99999999999999944e94 or 5.0000000000000001e-9 < k

   1. Initial program 75.4%

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

   2. Taylor expanded in k around 0 70.1%

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

   3. Taylor expanded in m around 0 23.9%

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

      1. +-commutative 23.9%

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

      2. *-commutative 23.9%

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

      3. fma-udef 23.9%

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

   5. Simplified 23.9%

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

   6. Taylor expanded in k around inf 23.9%

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

   if -8.99999999999999944e94 < k < 5.0000000000000001e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 37.2%

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

   3. Taylor expanded in k around 0 37.0%

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

   4. Taylor expanded in k around 0 36.3%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{+94} \lor \neg \left(k \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10: 45.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-299} \lor \neg \left(k \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.16e-299) (not (<= k 5e-9))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.16e-299) || !(k <= 5e-9)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 1.16d-299) .or. (.not. (k <= 5d-9))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.16e-299) || !(k <= 5e-9)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.16e-299) or not (k <= 5e-9):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.16e-299) || !(k <= 5e-9))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 1.16e-299) || ~((k <= 5e-9)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.16e-299], N[Not[LessEqual[k, 5e-9]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.15999999999999997e-299 or 5.0000000000000001e-9 < k

   1. Initial program 81.9%

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

   2. Taylor expanded in m around 0 42.1%

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

   3. Taylor expanded in k around inf 46.2%

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

      1. unpow2 46.2%

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

   5. Simplified 46.2%

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

   if 1.15999999999999997e-299 < k < 5.0000000000000001e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 53.7%

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

   3. Taylor expanded in k around 0 53.3%

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

   4. Taylor expanded in k around 0 53.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-299} \lor \neg \left(k \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 46.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-299}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-9}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.12e-299) (/ a (* k k)) (if (<= k 5e-9) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.12e-299) {
		tmp = a / (k * k);
	} else if (k <= 5e-9) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.12d-299) then
        tmp = a / (k * k)
    else if (k <= 5d-9) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.12e-299) {
		tmp = a / (k * k);
	} else if (k <= 5e-9) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.12e-299:
		tmp = a / (k * k)
	elif k <= 5e-9:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.12e-299)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 5e-9)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.12e-299)
		tmp = a / (k * k);
	elseif (k <= 5e-9)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.12e-299], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e-9], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.11999999999999998e-299

   1. Initial program 95.5%

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

   2. Taylor expanded in m around 0 23.6%

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

   3. Taylor expanded in k around inf 35.9%

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

      1. unpow2 35.9%

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

   5. Simplified 35.9%

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

   if 1.11999999999999998e-299 < k < 5.0000000000000001e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 53.7%

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

   3. Taylor expanded in k around 0 53.3%

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

   4. Taylor expanded in k around 0 53.3%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 73.1%

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

   2. Taylor expanded in m around 0 54.2%

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

   3. Taylor expanded in k around inf 52.9%

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

      1. unpow2 52.9%

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

   5. Simplified 52.9%

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

   6. Taylor expanded in a around 0 52.9%

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

      1. unpow2 52.9%

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

      2. associate-/r* 57.5%

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

   8. Simplified 57.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-299}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-9}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 12: 51.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+14) (/ a (* k k)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5.2d+14)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+14) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+14:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+14)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+14)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.2e14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.6%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

   if -5.2e14 < m

   1. Initial program 82.0%

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

   2. Taylor expanded in m around 0 51.2%

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

   3. Taylor expanded in k around 0 50.2%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 13: 19.6% 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. Taylor expanded in m around 0 46.0%

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

3. Taylor expanded in k around 0 45.4%

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

4. Taylor expanded in k around 0 20.7%

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

5. Final simplification 20.7%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023222 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))