Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 98.9%

Time: 9.9s

Alternatives: 16

Speedup: 1.1×

• 23.5% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -5.6 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 3.9:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -5.6e-18)
     t_0
     (if (<= m 2e-113)
       (/ 1.0 (+ (/ 1.0 a) (* (/ k a) (+ k 10.0))))
       (if (<= m 3.9) (/ (pow k m) (/ (+ 1.0 (* k (+ k 10.0))) a)) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -5.6e-18) {
		tmp = t_0;
	} else if (m <= 2e-113) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else if (m <= 3.9) {
		tmp = pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-5.6d-18)) then
        tmp = t_0
    else if (m <= 2d-113) then
        tmp = 1.0d0 / ((1.0d0 / a) + ((k / a) * (k + 10.0d0)))
    else if (m <= 3.9d0) then
        tmp = (k ** m) / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -5.6e-18) {
		tmp = t_0;
	} else if (m <= 2e-113) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else if (m <= 3.9) {
		tmp = Math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -5.6e-18:
		tmp = t_0
	elif m <= 2e-113:
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)))
	elif m <= 3.9:
		tmp = math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -5.6e-18)
		tmp = t_0;
	elseif (m <= 2e-113)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k / a) * Float64(k + 10.0))));
	elseif (m <= 3.9)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -5.6e-18)
		tmp = t_0;
	elseif (m <= 2e-113)
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	elseif (m <= 3.9)
		tmp = (k ^ m) / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.6e-18], t$95$0, If[LessEqual[m, 2e-113], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.9], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.60000000000000025e-18 or 3.89999999999999991 < m

   1. Initial program 87.0%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. associate-+l+ 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-rgt-out 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -5.60000000000000025e-18 < m < 1.99999999999999996e-113

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-/l* 93.7%

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

      3. sqr-neg 93.7%

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

      4. associate-+l+ 93.7%

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

      5. +-commutative 93.7%

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

      6. sqr-neg 93.7%

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

      7. distribute-rgt-out 93.7%

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

      8. fma-def 93.7%

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

      9. +-commutative 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in k around 0 93.7%

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

   5. Taylor expanded in m around 0 93.7%

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

      1. +-commutative 93.7%

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

      2. associate-+l+ 93.7%

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

      3. +-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. associate-*r/ 99.7%

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

      6. distribute-rgt-out 99.7%

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

   7. Simplified 99.7%

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

   if 1.99999999999999996e-113 < m < 3.89999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.8%

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

      3. sqr-neg 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. sqr-neg 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. fma-def 99.8%

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

      9. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.6 \cdot 10^{-18}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 3.9:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+231}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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+231) 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+231) {
		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+231) 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+231) {
		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+231:
		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+231)
		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+231)
		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+231], 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))) < 5.00000000000000028e231

   1. Initial program 97.7%

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

   if 5.00000000000000028e231 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 55.3%

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

      1. associate-/l* 55.3%

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

      2. sqr-neg 55.3%

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

      3. associate-+l+ 55.3%

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

      4. sqr-neg 55.3%

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

      5. distribute-rgt-out 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -5.6 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 3.9:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -5.6e-18)
     t_0
     (if (<= m 1.6e-111)
       (/ 1.0 (+ (/ 1.0 a) (* (/ k a) (+ k 10.0))))
       (if (<= m 3.9) (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m))) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -5.6e-18) {
		tmp = t_0;
	} else if (m <= 1.6e-111) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else if (m <= 3.9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-5.6d-18)) then
        tmp = t_0
    else if (m <= 1.6d-111) then
        tmp = 1.0d0 / ((1.0d0 / a) + ((k / a) * (k + 10.0d0)))
    else if (m <= 3.9d0) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -5.6e-18) {
		tmp = t_0;
	} else if (m <= 1.6e-111) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else if (m <= 3.9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -5.6e-18:
		tmp = t_0
	elif m <= 1.6e-111:
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)))
	elif m <= 3.9:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -5.6e-18)
		tmp = t_0;
	elseif (m <= 1.6e-111)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k / a) * Float64(k + 10.0))));
	elseif (m <= 3.9)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -5.6e-18)
		tmp = t_0;
	elseif (m <= 1.6e-111)
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	elseif (m <= 3.9)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.6e-18], t$95$0, If[LessEqual[m, 1.6e-111], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.9], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.60000000000000025e-18 or 3.89999999999999991 < m

   1. Initial program 87.0%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. associate-+l+ 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-rgt-out 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -5.60000000000000025e-18 < m < 1.5999999999999999e-111

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-/l* 93.7%

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

      3. sqr-neg 93.7%

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

      4. associate-+l+ 93.7%

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

      5. +-commutative 93.7%

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

      6. sqr-neg 93.7%

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

      7. distribute-rgt-out 93.7%

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

      8. fma-def 93.7%

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

      9. +-commutative 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in k around 0 93.7%

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

   5. Taylor expanded in m around 0 93.7%

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

      1. +-commutative 93.7%

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

      2. associate-+l+ 93.7%

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

      3. +-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. associate-*r/ 99.7%

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

      6. distribute-rgt-out 99.7%

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

   7. Simplified 99.7%

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

   if 1.5999999999999999e-111 < m < 3.89999999999999991

   1. Initial program 99.9%

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

      1. associate-/l* 99.7%

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

      2. sqr-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. sqr-neg 99.7%

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

      5. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.6 \cdot 10^{-18}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 3.9:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.6 \cdot 10^{-18} \lor \neg \left(m \leq 0.09\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.6e-18) (not (<= m 0.09)))
   (* a (pow k m))
   (/ 1.0 (+ (/ 1.0 a) (* (/ k a) (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.6e-18) || !(m <= 0.09)) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (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 <= (-5.6d-18)) .or. (.not. (m <= 0.09d0))) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / ((1.0d0 / a) + ((k / a) * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.6e-18) || !(m <= 0.09)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -5.6e-18) or not (m <= 0.09):
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5.6e-18) || !(m <= 0.09))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k / a) * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -5.6e-18) || ~((m <= 0.09)))
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5.6e-18], N[Not[LessEqual[m, 0.09]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.60000000000000025e-18 or 0.089999999999999997 < m

   1. Initial program 87.0%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. associate-+l+ 87.0%

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

      4. sqr-neg 87.0%

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

      5. distribute-rgt-out 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -5.60000000000000025e-18 < m < 0.089999999999999997

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/l* 94.9%

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

      3. sqr-neg 94.9%

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

      4. associate-+l+ 94.9%

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

      5. +-commutative 94.9%

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

      6. sqr-neg 94.9%

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

      7. distribute-rgt-out 94.9%

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

      8. fma-def 94.9%

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

      9. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in k around 0 94.9%

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

   5. Taylor expanded in m around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. associate-+l+ 93.2%

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

      3. +-commutative 93.2%

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

      4. unpow2 93.2%

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

      5. associate-*r/ 98.1%

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

      6. distribute-rgt-out 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.6 \cdot 10^{-18} \lor \neg \left(m \leq 0.09\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5: 54.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.84:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.058:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(\frac{12}{k} - k\right) + -5}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.84)
   (/ (/ a k) k)
   (if (<= m 0.058)
     (/ 1.0 (+ (/ 1.0 a) (* (/ k a) (+ k 10.0))))
     (/ (/ a (+ (- (/ 12.0 k) k) -5.0)) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.84) {
		tmp = (a / k) / k;
	} else if (m <= 0.058) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else {
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.84d0)) then
        tmp = (a / k) / k
    else if (m <= 0.058d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + ((k / a) * (k + 10.0d0)))
    else
        tmp = (a / (((12.0d0 / k) - k) + (-5.0d0))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.84) {
		tmp = (a / k) / k;
	} else if (m <= 0.058) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else {
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.84:
		tmp = (a / k) / k
	elif m <= 0.058:
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)))
	else:
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.84)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 0.058)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k / a) * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(a / Float64(Float64(Float64(12.0 / k) - k) + -5.0)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.84)
		tmp = (a / k) / k;
	elseif (m <= 0.058)
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	else
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.84], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.058], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(N[(12.0 / k), $MachinePrecision] - k), $MachinePrecision] + -5.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.839999999999999969

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. sqr-neg 98.8%

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

      3. associate-+l+ 98.8%

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

      4. sqr-neg 98.8%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 35.8%

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

      1. *-un-lft-identity 35.8%

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

      2. add-sqr-sqrt 35.8%

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

      3. times-frac 35.8%

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

      4. +-commutative 35.8%

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

      5. fma-def 35.8%

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

      6. +-commutative 35.8%

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

      7. +-commutative 35.8%

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

      8. fma-def 35.8%

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

      9. +-commutative 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*l/ 35.8%

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

      2. *-lft-identity 35.8%

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

      3. +-commutative 35.8%

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

      4. +-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in k around inf 41.1%

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

   10. Taylor expanded in k around inf 45.0%

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

   if -0.839999999999999969 < m < 0.0580000000000000029

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/l* 94.9%

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

      3. sqr-neg 94.9%

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

      4. associate-+l+ 94.9%

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

      5. +-commutative 94.9%

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

      6. sqr-neg 94.9%

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

      7. distribute-rgt-out 94.9%

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

      8. fma-def 94.9%

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

      9. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in k around 0 94.9%

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

   5. Taylor expanded in m around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. associate-+l+ 93.1%

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

      3. +-commutative 93.1%

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

      4. unpow2 93.1%

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

      5. associate-*r/ 98.0%

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

      6. distribute-rgt-out 98.0%

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

   7. Simplified 98.0%

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

   if 0.0580000000000000029 < m

   1. Initial program 74.4%

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

      1. associate-/l* 74.4%

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

      2. sqr-neg 74.4%

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

      3. associate-+l+ 74.4%

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

      4. sqr-neg 74.4%

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

      5. distribute-rgt-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in m around 0 4.3%

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

      1. *-un-lft-identity 4.3%

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

      2. add-sqr-sqrt 4.3%

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

      3. times-frac 4.3%

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

      4. +-commutative 4.3%

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

      5. fma-def 4.3%

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

      6. +-commutative 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-def 4.3%

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

      9. +-commutative 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. +-commutative 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in k around inf 3.4%

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

   10. Taylor expanded in k around -inf 15.9%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(-1 \cdot k + 12 \cdot \frac{1}{k}\right) - 5}}}{k} \]
   11. Step-by-step derivation

       1. sub-neg 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(-1 \cdot k + 12 \cdot \frac{1}{k}\right) + \left(-5\right)}}}{k} \]

       2. neg-mul-1 15.9%

          \[\leadsto \frac{\frac{a}{\left(\color{blue}{\left(-k\right)} + 12 \cdot \frac{1}{k}\right) + \left(-5\right)}}{k} \]

       3. +-commutative 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(12 \cdot \frac{1}{k} + \left(-k\right)\right)} + \left(-5\right)}}{k} \]

       4. unsub-neg 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(12 \cdot \frac{1}{k} - k\right)} + \left(-5\right)}}{k} \]

       5. associate-*r/ 15.9%

          \[\leadsto \frac{\frac{a}{\left(\color{blue}{\frac{12 \cdot 1}{k}} - k\right) + \left(-5\right)}}{k} \]

       6. metadata-eval 15.9%

          \[\leadsto \frac{\frac{a}{\left(\frac{\color{blue}{12}}{k} - k\right) + \left(-5\right)}}{k} \]

       7. metadata-eval 15.9%

          \[\leadsto \frac{\frac{a}{\left(\frac{12}{k} - k\right) + \color{blue}{-5}}}{k} \]

   12. Simplified 15.9%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(\frac{12}{k} - k\right) + -5}}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.84:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.058:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(\frac{12}{k} - k\right) + -5}}{k}\\ \end{array} \]

Alternative 6: 54.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.95:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(k + 5\right) + 12 \cdot \frac{-1}{k}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.95)
   (/ (/ a k) k)
   (if (<= m 0.052)
     (/ 1.0 (+ (/ 1.0 a) (* (/ k a) (+ k 10.0))))
     (/ (/ a (+ (+ k 5.0) (* 12.0 (/ -1.0 k)))) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.95) {
		tmp = (a / k) / k;
	} else if (m <= 0.052) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else {
		tmp = (a / ((k + 5.0) + (12.0 * (-1.0 / k)))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.95d0)) then
        tmp = (a / k) / k
    else if (m <= 0.052d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + ((k / a) * (k + 10.0d0)))
    else
        tmp = (a / ((k + 5.0d0) + (12.0d0 * ((-1.0d0) / k)))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.95) {
		tmp = (a / k) / k;
	} else if (m <= 0.052) {
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	} else {
		tmp = (a / ((k + 5.0) + (12.0 * (-1.0 / k)))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.95:
		tmp = (a / k) / k
	elif m <= 0.052:
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)))
	else:
		tmp = (a / ((k + 5.0) + (12.0 * (-1.0 / k)))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.95)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 0.052)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k / a) * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(a / Float64(Float64(k + 5.0) + Float64(12.0 * Float64(-1.0 / k)))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.95)
		tmp = (a / k) / k;
	elseif (m <= 0.052)
		tmp = 1.0 / ((1.0 / a) + ((k / a) * (k + 10.0)));
	else
		tmp = (a / ((k + 5.0) + (12.0 * (-1.0 / k)))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.95], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.052], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(k + 5.0), $MachinePrecision] + N[(12.0 * N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.94999999999999996

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. sqr-neg 98.8%

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

      3. associate-+l+ 98.8%

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

      4. sqr-neg 98.8%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 35.8%

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

      1. *-un-lft-identity 35.8%

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

      2. add-sqr-sqrt 35.8%

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

      3. times-frac 35.8%

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

      4. +-commutative 35.8%

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

      5. fma-def 35.8%

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

      6. +-commutative 35.8%

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

      7. +-commutative 35.8%

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

      8. fma-def 35.8%

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

      9. +-commutative 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*l/ 35.8%

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

      2. *-lft-identity 35.8%

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

      3. +-commutative 35.8%

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

      4. +-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in k around inf 41.1%

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

   10. Taylor expanded in k around inf 45.0%

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

   if -0.94999999999999996 < m < 0.0519999999999999976

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/l* 94.9%

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

      3. sqr-neg 94.9%

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

      4. associate-+l+ 94.9%

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

      5. +-commutative 94.9%

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

      6. sqr-neg 94.9%

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

      7. distribute-rgt-out 94.9%

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

      8. fma-def 94.9%

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

      9. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in k around 0 94.9%

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

   5. Taylor expanded in m around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. associate-+l+ 93.1%

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

      3. +-commutative 93.1%

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

      4. unpow2 93.1%

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

      5. associate-*r/ 98.0%

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

      6. distribute-rgt-out 98.0%

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

   7. Simplified 98.0%

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

   if 0.0519999999999999976 < m

   1. Initial program 74.4%

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

      1. associate-/l* 74.4%

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

      2. sqr-neg 74.4%

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

      3. associate-+l+ 74.4%

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

      4. sqr-neg 74.4%

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

      5. distribute-rgt-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in m around 0 4.3%

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

      1. *-un-lft-identity 4.3%

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

      2. add-sqr-sqrt 4.3%

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

      3. times-frac 4.3%

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

      4. +-commutative 4.3%

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

      5. fma-def 4.3%

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

      6. +-commutative 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-def 4.3%

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

      9. +-commutative 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. +-commutative 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in k around inf 3.4%

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

   10. Taylor expanded in k around inf 16.2%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(5 + k\right) - 12 \cdot \frac{1}{k}}}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.95:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(k + 5\right) + 12 \cdot \frac{-1}{k}}}{k}\\ \end{array} \]

Alternative 7: 52.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.11:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.055:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(\frac{12}{k} - k\right) + -5}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.11)
   (/ (/ a k) k)
   (if (<= m 0.055)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ (/ a (+ (- (/ 12.0 k) k) -5.0)) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.11) {
		tmp = (a / k) / k;
	} else if (m <= 0.055) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.11d0)) then
        tmp = (a / k) / k
    else if (m <= 0.055d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (a / (((12.0d0 / k) - k) + (-5.0d0))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.11) {
		tmp = (a / k) / k;
	} else if (m <= 0.055) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.11:
		tmp = (a / k) / k
	elif m <= 0.055:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.11)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 0.055)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(a / Float64(Float64(Float64(12.0 / k) - k) + -5.0)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.11)
		tmp = (a / k) / k;
	elseif (m <= 0.055)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = (a / (((12.0 / k) - k) + -5.0)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.11], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.055], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(N[(12.0 / k), $MachinePrecision] - k), $MachinePrecision] + -5.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.110000000000000001

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. sqr-neg 98.8%

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

      3. associate-+l+ 98.8%

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

      4. sqr-neg 98.8%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 35.8%

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

      1. *-un-lft-identity 35.8%

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

      2. add-sqr-sqrt 35.8%

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

      3. times-frac 35.8%

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

      4. +-commutative 35.8%

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

      5. fma-def 35.8%

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

      6. +-commutative 35.8%

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

      7. +-commutative 35.8%

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

      8. fma-def 35.8%

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

      9. +-commutative 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*l/ 35.8%

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

      2. *-lft-identity 35.8%

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

      3. +-commutative 35.8%

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

      4. +-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in k around inf 41.1%

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

   10. Taylor expanded in k around inf 45.0%

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

   if -0.110000000000000001 < m < 0.0550000000000000003

   1. Initial program 95.0%

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

      1. associate-/l* 95.0%

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

      2. sqr-neg 95.0%

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

      3. associate-+l+ 95.0%

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

      4. sqr-neg 95.0%

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

      5. distribute-rgt-out 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in m around 0 93.3%

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

   if 0.0550000000000000003 < m

   1. Initial program 74.4%

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

      1. associate-/l* 74.4%

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

      2. sqr-neg 74.4%

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

      3. associate-+l+ 74.4%

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

      4. sqr-neg 74.4%

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

      5. distribute-rgt-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in m around 0 4.3%

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

      1. *-un-lft-identity 4.3%

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

      2. add-sqr-sqrt 4.3%

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

      3. times-frac 4.3%

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

      4. +-commutative 4.3%

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

      5. fma-def 4.3%

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

      6. +-commutative 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-def 4.3%

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

      9. +-commutative 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. +-commutative 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in k around inf 3.4%

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

   10. Taylor expanded in k around -inf 15.9%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(-1 \cdot k + 12 \cdot \frac{1}{k}\right) - 5}}}{k} \]
   11. Step-by-step derivation

       1. sub-neg 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(-1 \cdot k + 12 \cdot \frac{1}{k}\right) + \left(-5\right)}}}{k} \]

       2. neg-mul-1 15.9%

          \[\leadsto \frac{\frac{a}{\left(\color{blue}{\left(-k\right)} + 12 \cdot \frac{1}{k}\right) + \left(-5\right)}}{k} \]

       3. +-commutative 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(12 \cdot \frac{1}{k} + \left(-k\right)\right)} + \left(-5\right)}}{k} \]

       4. unsub-neg 15.9%

          \[\leadsto \frac{\frac{a}{\color{blue}{\left(12 \cdot \frac{1}{k} - k\right)} + \left(-5\right)}}{k} \]

       5. associate-*r/ 15.9%

          \[\leadsto \frac{\frac{a}{\left(\color{blue}{\frac{12 \cdot 1}{k}} - k\right) + \left(-5\right)}}{k} \]

       6. metadata-eval 15.9%

          \[\leadsto \frac{\frac{a}{\left(\frac{\color{blue}{12}}{k} - k\right) + \left(-5\right)}}{k} \]

       7. metadata-eval 15.9%

          \[\leadsto \frac{\frac{a}{\left(\frac{12}{k} - k\right) + \color{blue}{-5}}}{k} \]

   12. Simplified 15.9%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(\frac{12}{k} - k\right) + -5}}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.11:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.055:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(\frac{12}{k} - k\right) + -5}}{k}\\ \end{array} \]

Alternative 8: 46.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2e-309)
   (/ (/ a k) k)
   (if (<= k 1.0) (/ a (+ 1.0 (* k 10.0))) (/ 1.0 (* (/ k a) (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2e-309) {
		tmp = (a / k) / k;
	} else if (k <= 1.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / ((k / a) * (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 (k <= 2d-309) then
        tmp = (a / k) / k
    else if (k <= 1.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = 1.0d0 / ((k / a) * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2e-309) {
		tmp = (a / k) / k;
	} else if (k <= 1.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / ((k / a) * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2e-309:
		tmp = (a / k) / k
	elif k <= 1.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = 1.0 / ((k / a) * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2e-309)
		tmp = Float64(Float64(a / k) / k);
	elseif (k <= 1.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(k / a) * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2e-309)
		tmp = (a / k) / k;
	elseif (k <= 1.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = 1.0 / ((k / a) * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2e-309], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 1.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999988e-309

   1. Initial program 91.3%

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

      1. associate-/l* 91.3%

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

      2. sqr-neg 91.3%

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

      3. associate-+l+ 91.3%

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

      4. sqr-neg 91.3%

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

      5. distribute-rgt-out 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in m around 0 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. add-sqr-sqrt 23.1%

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

      3. times-frac 23.1%

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

      4. +-commutative 23.1%

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

      5. fma-def 23.1%

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

      6. +-commutative 23.1%

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

      7. +-commutative 23.1%

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

      8. fma-def 23.1%

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

      9. +-commutative 23.1%

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

   6. Applied egg-rr 23.1%

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

      1. associate-*l/ 23.1%

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

      2. *-lft-identity 23.1%

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

      3. +-commutative 23.1%

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

      4. +-commutative 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in k around inf 21.9%

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

   10. Taylor expanded in k around inf 24.1%

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

   if 1.9999999999999988e-309 < k < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.5%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

   if 1 < k

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/l* 75.4%

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

      3. sqr-neg 75.4%

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

      4. associate-+l+ 75.4%

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

      5. +-commutative 75.4%

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

      6. sqr-neg 75.4%

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

      7. distribute-rgt-out 75.4%

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

      8. fma-def 75.4%

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

      9. +-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in k around 0 75.4%

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

   5. Taylor expanded in m around 0 57.2%

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

      1. +-commutative 57.2%

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

      2. associate-+l+ 57.1%

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

      3. +-commutative 57.1%

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

      4. unpow2 57.1%

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

      5. associate-*r/ 60.3%

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

      6. distribute-rgt-out 60.3%

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

   7. Simplified 60.3%

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

   8. Taylor expanded in a around 0 57.2%

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

   9. Taylor expanded in k around inf 56.3%

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

       1. unpow2 56.3%

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

       2. associate-*r/ 59.5%

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

       3. distribute-rgt-out 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9: 46.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2e-310) (not (<= k 0.1)))
   (/ (/ a k) k)
   (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2e-310) || !(k <= 0.1)) {
		tmp = (a / k) / k;
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-2d-310)) .or. (.not. (k <= 0.1d0))) then
        tmp = (a / k) / k
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2e-310) || !(k <= 0.1)) {
		tmp = (a / k) / k;
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2e-310) or not (k <= 0.1):
		tmp = (a / k) / k
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2e-310) || !(k <= 0.1))
		tmp = Float64(Float64(a / k) / k);
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -2e-310) || ~((k <= 0.1)))
		tmp = (a / k) / k;
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2e-310], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -1.999999999999994e-310 or 0.10000000000000001 < k

   1. Initial program 83.6%

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

      1. associate-/l* 83.6%

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

      2. sqr-neg 83.6%

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

      3. associate-+l+ 83.6%

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

      4. sqr-neg 83.6%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in m around 0 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. times-frac 41.9%

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

      4. +-commutative 41.9%

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

      5. fma-def 41.9%

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

      6. +-commutative 41.9%

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

      7. +-commutative 41.9%

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

      8. fma-def 41.9%

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

      9. +-commutative 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. associate-*l/ 41.9%

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

      2. *-lft-identity 41.9%

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

      3. +-commutative 41.9%

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

      4. +-commutative 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in k around inf 40.4%

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

   10. Taylor expanded in k around inf 42.9%

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

   if -1.999999999999994e-310 < 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. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.5%

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

   5. Taylor expanded in k around 0 55.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 10: 46.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309} \lor \neg \left(k \leq 10.2\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 3e-309) (not (<= k 10.2)))
   (/ (/ a k) k)
   (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 3e-309) || !(k <= 10.2)) {
		tmp = (a / k) / k;
	} else {
		tmp = a / (1.0 + (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 ((k <= 3d-309) .or. (.not. (k <= 10.2d0))) then
        tmp = (a / k) / k
    else
        tmp = a / (1.0d0 + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 3e-309) || !(k <= 10.2)) {
		tmp = (a / k) / k;
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 3e-309) or not (k <= 10.2):
		tmp = (a / k) / k
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 3e-309) || !(k <= 10.2))
		tmp = Float64(Float64(a / k) / k);
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 3e-309) || ~((k <= 10.2)))
		tmp = (a / k) / k;
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 3e-309], N[Not[LessEqual[k, 10.2]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.000000000000001e-309 or 10.199999999999999 < k

   1. Initial program 83.6%

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

      1. associate-/l* 83.6%

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

      2. sqr-neg 83.6%

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

      3. associate-+l+ 83.6%

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

      4. sqr-neg 83.6%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in m around 0 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. times-frac 41.9%

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

      4. +-commutative 41.9%

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

      5. fma-def 41.9%

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

      6. +-commutative 41.9%

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

      7. +-commutative 41.9%

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

      8. fma-def 41.9%

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

      9. +-commutative 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. associate-*l/ 41.9%

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

      2. *-lft-identity 41.9%

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

      3. +-commutative 41.9%

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

      4. +-commutative 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in k around inf 40.4%

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

   10. Taylor expanded in k around inf 42.9%

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

   if 3.000000000000001e-309 < k < 10.199999999999999

   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. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.5%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309} \lor \neg \left(k \leq 10.2\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 11: 46.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 5.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k + 5}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2e-310)
   (/ (/ a k) k)
   (if (<= k 5.2) (/ a (+ 1.0 (* k 10.0))) (/ (/ a (+ k 5.0)) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2e-310) {
		tmp = (a / k) / k;
	} else if (k <= 5.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / (k + 5.0)) / 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 <= (-2d-310)) then
        tmp = (a / k) / k
    else if (k <= 5.2d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / (k + 5.0d0)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2e-310) {
		tmp = (a / k) / k;
	} else if (k <= 5.2) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / (k + 5.0)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2e-310:
		tmp = (a / k) / k
	elif k <= 5.2:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / (k + 5.0)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2e-310)
		tmp = Float64(Float64(a / k) / k);
	elseif (k <= 5.2)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / Float64(k + 5.0)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2e-310)
		tmp = (a / k) / k;
	elseif (k <= 5.2)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / (k + 5.0)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2e-310], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 5.2], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(k + 5.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.999999999999994e-310

   1. Initial program 91.3%

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

      1. associate-/l* 91.3%

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

      2. sqr-neg 91.3%

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

      3. associate-+l+ 91.3%

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

      4. sqr-neg 91.3%

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

      5. distribute-rgt-out 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in m around 0 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. add-sqr-sqrt 23.1%

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

      3. times-frac 23.1%

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

      4. +-commutative 23.1%

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

      5. fma-def 23.1%

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

      6. +-commutative 23.1%

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

      7. +-commutative 23.1%

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

      8. fma-def 23.1%

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

      9. +-commutative 23.1%

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

   6. Applied egg-rr 23.1%

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

      1. associate-*l/ 23.1%

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

      2. *-lft-identity 23.1%

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

      3. +-commutative 23.1%

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

      4. +-commutative 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in k around inf 21.9%

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

   10. Taylor expanded in k around inf 24.1%

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

   if -1.999999999999994e-310 < k < 5.20000000000000018

   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. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.5%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

   if 5.20000000000000018 < k

   1. Initial program 77.6%

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

      1. associate-/l* 77.6%

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

      2. sqr-neg 77.6%

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

      3. associate-+l+ 77.6%

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

      4. sqr-neg 77.6%

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

      5. distribute-rgt-out 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in m around 0 56.6%

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

      1. *-un-lft-identity 56.6%

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

      2. add-sqr-sqrt 56.6%

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

      3. times-frac 56.6%

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

      4. +-commutative 56.6%

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

      5. fma-def 56.6%

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

      6. +-commutative 56.6%

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

      7. +-commutative 56.6%

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

      8. fma-def 56.6%

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

      9. +-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 56.6%

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

      2. *-lft-identity 56.6%

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

      3. +-commutative 56.6%

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

      4. +-commutative 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in k around inf 54.9%

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

   10. Taylor expanded in k around inf 57.7%

       \[\leadsto \frac{\frac{a}{\color{blue}{5 + k}}}{k} \]
   11. Step-by-step derivation

       1. +-commutative 57.7%

          \[\leadsto \frac{\frac{a}{\color{blue}{k + 5}}}{k} \]

   12. Simplified 57.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 5.2:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k + 5}}{k}\\ \end{array} \]

Alternative 12: 48.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.055:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.055) (/ (/ a k) k) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.055) {
		tmp = (a / k) / k;
	} 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 <= (-0.055d0)) then
        tmp = (a / k) / k
    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 <= -0.055) {
		tmp = (a / k) / k;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.055:
		tmp = (a / k) / k
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.055)
		tmp = Float64(Float64(a / k) / k);
	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 <= -0.055)
		tmp = (a / k) / k;
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.055], N[(N[(a / k), $MachinePrecision] / k), $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 < -0.0550000000000000003

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. sqr-neg 98.8%

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

      3. associate-+l+ 98.8%

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

      4. sqr-neg 98.8%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 35.8%

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

      1. *-un-lft-identity 35.8%

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

      2. add-sqr-sqrt 35.8%

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

      3. times-frac 35.8%

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

      4. +-commutative 35.8%

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

      5. fma-def 35.8%

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

      6. +-commutative 35.8%

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

      7. +-commutative 35.8%

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

      8. fma-def 35.8%

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

      9. +-commutative 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*l/ 35.8%

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

      2. *-lft-identity 35.8%

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

      3. +-commutative 35.8%

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

      4. +-commutative 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in k around inf 41.1%

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

   10. Taylor expanded in k around inf 45.0%

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

   if -0.0550000000000000003 < m

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

      2. sqr-neg 85.7%

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

      3. associate-+l+ 85.7%

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

      4. sqr-neg 85.7%

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

      5. distribute-rgt-out 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in m around 0 53.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.055:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 13: 46.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 3e-309) (not (<= k 1.0))) (/ (/ a k) k) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 3e-309) || !(k <= 1.0)) {
		tmp = (a / k) / k;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 3d-309) .or. (.not. (k <= 1.0d0))) then
        tmp = (a / k) / k
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 3e-309) || !(k <= 1.0)) {
		tmp = (a / k) / k;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 3e-309) or not (k <= 1.0):
		tmp = (a / k) / k
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 3e-309) || !(k <= 1.0))
		tmp = Float64(Float64(a / k) / k);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 3e-309) || ~((k <= 1.0)))
		tmp = (a / k) / k;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 3e-309], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 3.000000000000001e-309 or 1 < k

   1. Initial program 83.6%

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

      1. associate-/l* 83.6%

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

      2. sqr-neg 83.6%

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

      3. associate-+l+ 83.6%

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

      4. sqr-neg 83.6%

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

      5. distribute-rgt-out 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in m around 0 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. times-frac 41.9%

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

      4. +-commutative 41.9%

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

      5. fma-def 41.9%

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

      6. +-commutative 41.9%

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

      7. +-commutative 41.9%

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

      8. fma-def 41.9%

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

      9. +-commutative 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. associate-*l/ 41.9%

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

      2. *-lft-identity 41.9%

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

      3. +-commutative 41.9%

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

      4. +-commutative 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in k around inf 40.4%

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

   10. Taylor expanded in k around inf 42.9%

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

   if 3.000000000000001e-309 < k < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.5%

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

   5. Taylor expanded in k around 0 54.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 14: 26.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 96.4%

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

      1. associate-/l* 96.4%

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

      2. sqr-neg 96.4%

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

      3. associate-+l+ 96.4%

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

      4. sqr-neg 96.4%

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

      5. distribute-rgt-out 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in m around 0 42.8%

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

   5. Taylor expanded in k around 0 33.3%

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

   if 0.10000000000000001 < k

   1. Initial program 77.6%

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

      1. associate-/l* 77.6%

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

      2. sqr-neg 77.6%

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

      3. associate-+l+ 77.6%

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

      4. sqr-neg 77.6%

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

      5. distribute-rgt-out 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in m around 0 56.6%

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

   5. Taylor expanded in k around 0 23.5%

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

      1. *-commutative 23.5%

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

   7. Simplified 23.5%

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

   8. Taylor expanded in k around inf 23.5%

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

4. Final simplification 30.0%

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

Alternative 15: 26.3% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k 1.0) a (/ a k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = a / k;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 96.4%

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

      1. associate-/l* 96.4%

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

      2. sqr-neg 96.4%

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

      3. associate-+l+ 96.4%

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

      4. sqr-neg 96.4%

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

      5. distribute-rgt-out 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in m around 0 42.8%

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

   5. Taylor expanded in k around 0 33.3%

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

   if 1 < k

   1. Initial program 77.6%

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

      1. associate-/l* 77.6%

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

      2. sqr-neg 77.6%

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

      3. associate-+l+ 77.6%

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

      4. sqr-neg 77.6%

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

      5. distribute-rgt-out 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in m around 0 56.6%

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

      1. *-un-lft-identity 56.6%

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

      2. add-sqr-sqrt 56.6%

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

      3. times-frac 56.6%

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

      4. +-commutative 56.6%

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

      5. fma-def 56.6%

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

      6. +-commutative 56.6%

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

      7. +-commutative 56.6%

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

      8. fma-def 56.6%

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

      9. +-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 56.6%

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

      2. *-lft-identity 56.6%

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

      3. +-commutative 56.6%

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

      4. +-commutative 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in k around inf 54.9%

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

   10. Taylor expanded in k around 0 23.4%

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

4. Final simplification 29.9%

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

Alternative 16: 20.0% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.0%

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

   1. associate-/l* 89.9%

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

   2. sqr-neg 89.9%

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

   3. associate-+l+ 89.9%

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

   4. sqr-neg 89.9%

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

   5. distribute-rgt-out 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in m around 0 47.5%

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

5. Taylor expanded in k around 0 23.4%

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

6. Final simplification 23.4%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023334 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))