Falkner and Boettcher, Appendix A

Percentage Accurate: 90.9% → 96.8%

Time: 8.3s

Alternatives: 14

Speedup: 1.1×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e-34)
   (* (/ (pow k m) k) (/ a k))
   (if (<= m 5e-8) (/ a (fma k (+ k 10.0) 1.0)) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e-34) {
		tmp = (pow(k, m) / k) * (a / k);
	} else if (m <= 5e-8) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e-34)
		tmp = Float64(Float64((k ^ m) / k) * Float64(a / k));
	elseif (m <= 5e-8)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e-34], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-8], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e-34

   1. Initial program 98.1%

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

      1. associate-/l* 98.1%

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

      2. associate-+l+ 98.1%

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

      3. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in k around inf 63.0%

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

      1. unpow2 63.0%

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

      2. associate-/l* 63.0%

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

      3. mul-1-neg 63.0%

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

      4. exp-neg 63.0%

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

      5. log-rec 63.0%

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

      6. distribute-lft-neg-in 63.0%

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

      7. rec-exp 63.0%

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

      8. exp-to-pow 97.1%

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

   6. Simplified 97.1%

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

      1. associate-/r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. remove-double-div 99.0%

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

   8. Applied egg-rr 99.0%

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

   if -1.05e-34 < m < 4.9999999999999998e-8

   1. Initial program 95.7%

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

      1. associate-/l* 95.7%

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

      2. associate-+l+ 95.7%

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

      3. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 94.6%

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

      1. +-commutative 94.6%

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

      2. unpow2 94.6%

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

      3. distribute-rgt-in 94.6%

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

      4. fma-udef 94.6%

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

   6. Simplified 94.6%

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

   if 4.9999999999999998e-8 < m

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*r/ 65.8%

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

      3. associate-+l+ 65.8%

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

      4. +-commutative 65.8%

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

      5. distribute-rgt-out 65.8%

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

      6. fma-def 65.8%

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

      7. +-commutative 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+164)
     (* (pow k m) (/ a (fma k (+ k 10.0) 1.0)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+164) {
		tmp = pow(k, m) * (a / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+164)
		tmp = Float64((k ^ m) * Float64(a / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+164], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 1e164

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*r/ 97.4%

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

      3. associate-+l+ 97.4%

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

      4. +-commutative 97.4%

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

      5. distribute-rgt-out 97.4%

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

      6. fma-def 97.4%

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

      7. +-commutative 97.4%

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

   3. Simplified 97.4%

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

   if 1e164 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 58.5%

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

      1. *-commutative 58.5%

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

      2. associate-*r/ 49.1%

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

      3. associate-+l+ 49.1%

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

      4. +-commutative 49.1%

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

      5. distribute-rgt-out 49.1%

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

      6. fma-def 49.1%

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

      7. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+164)
     (/ a (/ (+ 1.0 (+ (* k 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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+164) {
		tmp = a / ((1.0 + ((k * 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 ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 1d+164) then
        tmp = a / ((1.0d0 + ((k * 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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+164) {
		tmp = a / ((1.0 + ((k * 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 (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+164:
		tmp = a / ((1.0 + ((k * 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 (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+164)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(Float64(k * 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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+164)
		tmp = a / ((1.0 + ((k * 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[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+164], N[(a / N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 1e164

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-+l+ 97.4%

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

      3. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   if 1e164 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 58.5%

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

      1. *-commutative 58.5%

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

      2. associate-*r/ 49.1%

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

      3. associate-+l+ 49.1%

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

      4. +-commutative 49.1%

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

      5. distribute-rgt-out 49.1%

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

      6. fma-def 49.1%

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

      7. +-commutative 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(k / N[(N[(1.0 / N[(1.0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $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. *-commutative 96.4%

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

      2. associate-*r/ 94.6%

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

      3. associate-+l+ 94.6%

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

      4. +-commutative 94.6%

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

      5. distribute-rgt-out 94.6%

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

      6. fma-def 94.6%

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

      7. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in k around 0 98.7%

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

   if 1 < k

   1. Initial program 75.8%

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

      1. associate-/l* 75.8%

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

      2. associate-+l+ 75.8%

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

      3. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in k around inf 74.3%

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

      1. unpow2 74.3%

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

      2. associate-/l* 92.5%

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

      3. mul-1-neg 92.5%

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

      4. exp-neg 92.5%

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

      5. log-rec 92.5%

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

      6. distribute-lft-neg-in 92.5%

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

      7. rec-exp 92.5%

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

      8. exp-to-pow 92.5%

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

   6. Simplified 92.5%

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

4. Final simplification 96.6%

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

Alternative 5: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.025 \lor \neg \left(m \leq 8.2 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.025) (not (<= m 8.2e-8)))
   (* a (pow k m))
   (/ a (fma k (+ k 10.0) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.025) || !(m <= 8.2e-8)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / fma(k, (k + 10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.025) || !(m <= 8.2e-8))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.025], N[Not[LessEqual[m, 8.2e-8]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.025000000000000001 or 8.20000000000000063e-8 < m

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*r/ 84.1%

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

      3. associate-+l+ 84.1%

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

      4. +-commutative 84.1%

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

      5. distribute-rgt-out 84.1%

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

      6. fma-def 84.1%

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

      7. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -0.025000000000000001 < m < 8.20000000000000063e-8

   1. Initial program 93.8%

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

      1. associate-/l* 93.8%

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

      2. associate-+l+ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in m around 0 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. distribute-rgt-in 92.2%

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

      4. fma-udef 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.025 \lor \neg \left(m \leq 8.2 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \]

Alternative 6: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00025 \lor \neg \left(m \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.00025) (not (<= m 2.15e-7)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00025) || !(m <= 2.15e-7)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-0.00025d0)) .or. (.not. (m <= 2.15d-7))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00025) || !(m <= 2.15e-7)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.00025) or not (m <= 2.15e-7):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.00025) || !(m <= 2.15e-7))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.00025) || ~((m <= 2.15e-7)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.00025], N[Not[LessEqual[m, 2.15e-7]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.5000000000000001e-4 or 2.1500000000000001e-7 < m

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*r/ 84.1%

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

      3. associate-+l+ 84.1%

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

      4. +-commutative 84.1%

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

      5. distribute-rgt-out 84.1%

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

      6. fma-def 84.1%

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

      7. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -2.5000000000000001e-4 < m < 2.1500000000000001e-7

   1. Initial program 93.8%

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

      1. associate-/l* 93.8%

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

      2. associate-+l+ 93.8%

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

      3. *-commutative 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in m around 0 92.2%

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

      1. +-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. distribute-rgt-in 92.2%

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

      4. fma-udef 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in a around 0 92.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00025 \lor \neg \left(m \leq 2.15 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7: 45.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -5e-310)
     t_0
     (if (<= k 1.0) a (if (<= k 1.7e+196) t_0 (* (/ a k) (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -5e-310) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else if (k <= 1.7e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) * (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= (-5d-310)) then
        tmp = t_0
    else if (k <= 1.0d0) then
        tmp = a
    else if (k <= 1.7d+196) then
        tmp = t_0
    else
        tmp = (a / k) * (1.0d0 / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -5e-310) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else if (k <= 1.7e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= -5e-310:
		tmp = t_0
	elif k <= 1.0:
		tmp = a
	elif k <= 1.7e+196:
		tmp = t_0
	else:
		tmp = (a / k) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -5e-310)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	elseif (k <= 1.7e+196)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	elseif (k <= 1.7e+196)
		tmp = t_0;
	else
		tmp = (a / k) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -5e-310], t$95$0, If[LessEqual[k, 1.0], a, If[LessEqual[k, 1.7e+196], t$95$0, N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310 or 1 < k < 1.7e196

   1. Initial program 90.9%

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

      1. associate-/l* 91.0%

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

      2. associate-+l+ 91.0%

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

      3. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in k around inf 37.4%

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

      1. unpow2 37.4%

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

      2. associate-/l* 41.5%

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

      3. mul-1-neg 41.5%

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

      4. exp-neg 41.5%

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

      5. log-rec 41.5%

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

      6. distribute-lft-neg-in 41.5%

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

      7. rec-exp 41.5%

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

      8. exp-to-pow 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

   9. Simplified 40.6%

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

   if -4.999999999999985e-310 < 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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 97.8%

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

   5. Taylor expanded in m around 0 49.8%

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

   if 1.7e196 < k

   1. Initial program 54.9%

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

      1. associate-/l* 54.9%

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

      2. associate-+l+ 54.9%

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

      3. *-commutative 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

      2. associate-/l* 85.4%

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

      3. mul-1-neg 85.4%

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

      4. exp-neg 85.4%

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

      5. log-rec 85.4%

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

      6. distribute-lft-neg-in 85.4%

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

      7. rec-exp 85.4%

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

      8. exp-to-pow 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in m around 0 55.3%

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

      1. unpow2 55.3%

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

   9. Simplified 55.3%

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

       1. associate-/r* 66.9%

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

       2. div-inv 66.9%

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

   11. Applied egg-rr 66.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 8: 46.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.4 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.082:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 1.4e-302)
     t_0
     (if (<= k 0.082)
       (+ a (* -10.0 (* a k)))
       (if (<= k 1.55e+196) t_0 (* (/ a k) (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.4e-302) {
		tmp = t_0;
	} else if (k <= 0.082) {
		tmp = a + (-10.0 * (a * k));
	} else if (k <= 1.55e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) * (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.4d-302) then
        tmp = t_0
    else if (k <= 0.082d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else if (k <= 1.55d+196) then
        tmp = t_0
    else
        tmp = (a / k) * (1.0d0 / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.4e-302) {
		tmp = t_0;
	} else if (k <= 0.082) {
		tmp = a + (-10.0 * (a * k));
	} else if (k <= 1.55e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.4e-302:
		tmp = t_0
	elif k <= 0.082:
		tmp = a + (-10.0 * (a * k))
	elif k <= 1.55e+196:
		tmp = t_0
	else:
		tmp = (a / k) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.4e-302)
		tmp = t_0;
	elseif (k <= 0.082)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	elseif (k <= 1.55e+196)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 1.4e-302)
		tmp = t_0;
	elseif (k <= 0.082)
		tmp = a + (-10.0 * (a * k));
	elseif (k <= 1.55e+196)
		tmp = t_0;
	else
		tmp = (a / k) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.4e-302], t$95$0, If[LessEqual[k, 0.082], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+196], t$95$0, N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4e-302 or 0.0820000000000000034 < k < 1.55000000000000005e196

   1. Initial program 90.9%

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

      1. associate-/l* 91.0%

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

      2. associate-+l+ 91.0%

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

      3. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in k around inf 37.4%

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

      1. unpow2 37.4%

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

      2. associate-/l* 41.5%

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

      3. mul-1-neg 41.5%

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

      4. exp-neg 41.5%

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

      5. log-rec 41.5%

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

      6. distribute-lft-neg-in 41.5%

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

      7. rec-exp 41.5%

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

      8. exp-to-pow 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

   9. Simplified 40.6%

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

   if 1.4e-302 < k < 0.0820000000000000034

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 98.8%

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

      1. associate-*r* 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in m around 0 50.8%

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

   if 1.55000000000000005e196 < k

   1. Initial program 54.9%

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

      1. associate-/l* 54.9%

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

      2. associate-+l+ 54.9%

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

      3. *-commutative 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

      2. associate-/l* 85.4%

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

      3. mul-1-neg 85.4%

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

      4. exp-neg 85.4%

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

      5. log-rec 85.4%

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

      6. distribute-lft-neg-in 85.4%

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

      7. rec-exp 85.4%

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

      8. exp-to-pow 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in m around 0 55.3%

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

      1. unpow2 55.3%

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

   9. Simplified 55.3%

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

       1. associate-/r* 66.9%

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

       2. div-inv 66.9%

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

   11. Applied egg-rr 66.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.082:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 9: 45.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 6.6 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 6.6e-308)
     t_0
     (if (<= k 1.0) a (if (<= k 2.6e+196) t_0 (/ (/ a k) k))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 6.6e-308) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else if (k <= 2.6e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 6.6d-308) then
        tmp = t_0
    else if (k <= 1.0d0) then
        tmp = a
    else if (k <= 2.6d+196) then
        tmp = t_0
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 6.6e-308) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else if (k <= 2.6e+196) {
		tmp = t_0;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 6.6e-308:
		tmp = t_0
	elif k <= 1.0:
		tmp = a
	elif k <= 2.6e+196:
		tmp = t_0
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 6.6e-308)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	elseif (k <= 2.6e+196)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 6.6e-308)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	elseif (k <= 2.6e+196)
		tmp = t_0;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.6e-308], t$95$0, If[LessEqual[k, 1.0], a, If[LessEqual[k, 2.6e+196], t$95$0, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.5999999999999996e-308 or 1 < k < 2.60000000000000012e196

   1. Initial program 90.9%

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

      1. associate-/l* 91.0%

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

      2. associate-+l+ 91.0%

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

      3. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in k around inf 37.4%

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

      1. unpow2 37.4%

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

      2. associate-/l* 41.5%

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

      3. mul-1-neg 41.5%

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

      4. exp-neg 41.5%

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

      5. log-rec 41.5%

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

      6. distribute-lft-neg-in 41.5%

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

      7. rec-exp 41.5%

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

      8. exp-to-pow 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in m around 0 40.6%

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

      1. unpow2 40.6%

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

   9. Simplified 40.6%

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

   if 6.5999999999999996e-308 < 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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 97.8%

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

   5. Taylor expanded in m around 0 49.8%

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

   if 2.60000000000000012e196 < k

   1. Initial program 54.9%

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

      1. associate-/l* 54.9%

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

      2. associate-+l+ 54.9%

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

      3. *-commutative 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

      2. associate-/l* 85.4%

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

      3. mul-1-neg 85.4%

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

      4. exp-neg 85.4%

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

      5. log-rec 85.4%

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

      6. distribute-lft-neg-in 85.4%

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

      7. rec-exp 85.4%

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

      8. exp-to-pow 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in m around 0 55.3%

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

      1. unpow2 55.3%

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

   9. Simplified 55.3%

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

       1. associate-/r* 66.9%

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

       2. div-inv 66.9%

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

   11. Applied egg-rr 66.9%

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

       1. un-div-inv 66.9%

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

   13. Applied egg-rr 66.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10: 42.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-43)
   (/ a (* k k))
   (if (<= m 3.3e+31) (/ a (+ 1.0 (* k 10.0))) (+ a (* -10.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-43) {
		tmp = a / (k * k);
	} else if (m <= 3.3e+31) {
		tmp = a / (1.0 + (k * 10.0));
	} 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 (m <= (-5.4d-43)) then
        tmp = a / (k * k)
    else if (m <= 3.3d+31) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    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 (m <= -5.4e-43) {
		tmp = a / (k * k);
	} else if (m <= 3.3e+31) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.4e-43:
		tmp = a / (k * k)
	elif m <= 3.3e+31:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e-43)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 3.3e+31)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	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 (m <= -5.4e-43)
		tmp = a / (k * k);
	elseif (m <= 3.3e+31)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e-43], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.3e+31], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.39999999999999982e-43

   1. Initial program 98.1%

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

      1. associate-/l* 98.1%

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

      2. associate-+l+ 98.1%

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

      3. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in k around inf 62.9%

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

      1. unpow2 62.9%

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

      2. associate-/l* 62.8%

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

      3. mul-1-neg 62.8%

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

      4. exp-neg 62.8%

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

      5. log-rec 62.8%

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

      6. distribute-lft-neg-in 62.8%

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

      7. rec-exp 62.8%

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

      8. exp-to-pow 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in m around 0 64.0%

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

      1. unpow2 64.0%

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

   9. Simplified 64.0%

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

   if -5.39999999999999982e-43 < m < 3.29999999999999992e31

   1. Initial program 93.6%

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

      1. associate-/l* 93.6%

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

      2. associate-+l+ 93.6%

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

      3. *-commutative 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in m around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. unpow2 87.3%

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

      3. distribute-rgt-in 87.3%

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

      4. fma-udef 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in a around 0 87.3%

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

   8. Taylor expanded in k around 0 68.7%

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

      1. *-commutative 68.7%

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

   10. Simplified 68.7%

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

   if 3.29999999999999992e31 < m

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*r/ 65.3%

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

      3. associate-+l+ 65.3%

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

      4. +-commutative 65.3%

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

      5. distribute-rgt-out 65.3%

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

      6. fma-def 65.3%

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

      7. +-commutative 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in k around 0 76.4%

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

      1. associate-*r* 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in m around 0 8.4%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 51.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.14:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.14) (/ a (* k k)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.14) {
		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 <= (-0.14d0)) 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 <= -0.14) {
		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 <= -0.14:
		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 <= -0.14)
		tmp = Float64(a / Float64(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 <= -0.14)
		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, -0.14], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.14000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. unpow2 63.7%

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

      2. associate-/l* 63.7%

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

      3. mul-1-neg 63.7%

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

      4. exp-neg 63.7%

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

      5. log-rec 63.7%

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

      6. distribute-lft-neg-in 63.7%

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

      7. rec-exp 63.7%

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

      8. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 65.2%

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

      1. unpow2 65.2%

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

   9. Simplified 65.2%

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

   if -0.14000000000000001 < m

   1. Initial program 83.4%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.5%

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

      3. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in m around 0 49.7%

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

      1. +-commutative 49.7%

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

      2. unpow2 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. fma-udef 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in a around 0 49.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.14:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 12: 45.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-303} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.3e-303) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.3e-303) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 1.3d-303) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.3e-303) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.3e-303) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.3e-303) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 1.3e-303) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.3e-303], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.30000000000000002e-303 or 1 < k

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. associate-+l+ 82.7%

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

      3. *-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in k around inf 41.4%

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

      1. unpow2 41.4%

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

      2. associate-/l* 51.5%

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

      3. mul-1-neg 51.5%

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

      4. exp-neg 51.5%

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

      5. log-rec 51.5%

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

      6. distribute-lft-neg-in 51.5%

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

      7. rec-exp 51.5%

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

      8. exp-to-pow 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in m around 0 44.0%

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

      1. unpow2 44.0%

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

   9. Simplified 44.0%

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

   if 1.30000000000000002e-303 < 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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 97.8%

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

   5. Taylor expanded in m around 0 49.8%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-303} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 13: 50.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.6:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.6) (/ a (* k k)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.6) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.6d0)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.6) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.6:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.6)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.6)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.6], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.599999999999999978

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. unpow2 63.7%

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

      2. associate-/l* 63.7%

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

      3. mul-1-neg 63.7%

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

      4. exp-neg 63.7%

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

      5. log-rec 63.7%

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

      6. distribute-lft-neg-in 63.7%

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

      7. rec-exp 63.7%

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

      8. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 65.2%

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

      1. unpow2 65.2%

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

   9. Simplified 65.2%

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

   if -0.599999999999999978 < m

   1. Initial program 83.4%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.5%

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

      3. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in m around 0 49.7%

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

      1. +-commutative 49.7%

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

      2. unpow2 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. fma-udef 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in a around 0 49.7%

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

   8. Taylor expanded in k around inf 47.6%

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

      1. unpow2 47.6%

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

   10. Simplified 47.6%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.6:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 14: 19.0% 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 89.3%

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

   1. *-commutative 89.3%

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

   2. associate-*r/ 87.4%

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

   3. associate-+l+ 87.4%

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

   4. +-commutative 87.4%

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

   5. distribute-rgt-out 87.4%

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

   6. fma-def 87.4%

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

   7. +-commutative 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in k around 0 85.5%

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

5. Taylor expanded in m around 0 21.6%

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

6. Final simplification 21.6%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023185 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))