Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 98.6%

Time: 10.5s

Alternatives: 14

Speedup: 1.0×

• 21.9% 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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-23}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.0195:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.5e-23)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m 0.0195)
     (* (/ a (hypot 1.0 k)) (/ 1.0 (hypot 1.0 k)))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-23) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.0195) {
		tmp = (a / hypot(1.0, k)) * (1.0 / hypot(1.0, k));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-23) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.0195) {
		tmp = (a / Math.hypot(1.0, k)) * (1.0 / Math.hypot(1.0, k));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.5e-23:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	elif m <= 0.0195:
		tmp = (a / math.hypot(1.0, k)) * (1.0 / math.hypot(1.0, k))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.5e-23)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	elseif (m <= 0.0195)
		tmp = Float64(Float64(a / hypot(1.0, k)) * Float64(1.0 / hypot(1.0, k)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.5e-23)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	elseif (m <= 0.0195)
		tmp = (a / hypot(1.0, k)) * (1.0 / hypot(1.0, k));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.5e-23], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0195], N[(N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.49999999999999993e-23

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -3.49999999999999993e-23 < m < 0.0195

   1. Initial program 94.0%

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

      1. associate-/l* 94.0%

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

      2. remove-double-neg 94.0%

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

      3. distribute-frac-neg2 94.0%

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

      4. distribute-neg-frac2 94.0%

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

      5. remove-double-neg 94.0%

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

      6. sqr-neg 94.0%

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

      7. associate-+l+ 94.0%

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

      8. sqr-neg 94.0%

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

      9. distribute-rgt-out 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in m around 0 94.0%

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

   6. Taylor expanded in k around inf 93.1%

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

      1. *-rgt-identity 93.1%

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

      2. add-sqr-sqrt 93.1%

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

      3. times-frac 93.1%

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

      4. hypot-1-def 93.1%

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

      5. hypot-1-def 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 0.0195 < m

   1. Initial program 74.2%

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

      1. associate-/l* 74.1%

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

      2. remove-double-neg 74.1%

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

      3. distribute-frac-neg2 74.1%

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

      4. distribute-neg-frac2 74.1%

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

      5. remove-double-neg 74.1%

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

      6. sqr-neg 74.1%

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

      7. associate-+l+ 74.1%

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

      8. sqr-neg 74.1%

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

      9. distribute-rgt-out 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in k around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-23}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.0195:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k))) INFINITY)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (* a (+ 1.0 (* k (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= ((double) INFINITY)) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (((a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= Double.POSITIVE_INFINITY) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if ((a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))) <= math.inf:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * (1.0 + (k * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= Inf)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (((a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k))) <= Inf)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (1.0 + (k * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 97.6%

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

      1. associate-/l* 97.6%

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

      2. remove-double-neg 97.6%

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

      3. distribute-frac-neg2 97.6%

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

      4. distribute-neg-frac2 97.6%

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

      5. remove-double-neg 97.6%

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

      6. sqr-neg 97.6%

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

      7. associate-+l+ 97.6%

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

      8. sqr-neg 97.6%

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

      9. distribute-rgt-out 97.6%

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

   3. Simplified 97.6%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 0.0%

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

      1. associate-/l* 0.0%

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

      2. remove-double-neg 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. distribute-neg-frac2 0.0%

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

      5. remove-double-neg 0.0%

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

      6. sqr-neg 0.0%

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

      7. associate-+l+ 0.0%

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

      8. sqr-neg 0.0%

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

      9. distribute-rgt-out 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in m around 0 1.6%

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

   6. Taylor expanded in k around 0 100.0%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 100.0%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq \infty:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{+28} \lor \neg \left(m \leq 0.0142\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.55e+28) (not (<= m 0.0142)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.55e+28) || !(m <= 0.0142)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-1.55d+28)) .or. (.not. (m <= 0.0142d0))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -1.55e+28) || !(m <= 0.0142)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.55e+28) or not (m <= 0.0142):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.55e+28) || !(m <= 0.0142))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -1.55e+28) || ~((m <= 0.0142)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.55e+28], N[Not[LessEqual[m, 0.0142]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.55e28 or 0.014200000000000001 < m

   1. Initial program 85.1%

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

      1. associate-/l* 85.1%

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

      2. remove-double-neg 85.1%

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

      3. distribute-frac-neg2 85.1%

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

      4. distribute-neg-frac2 85.1%

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

      5. remove-double-neg 85.1%

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

      6. sqr-neg 85.1%

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

      7. associate-+l+ 85.1%

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

      8. sqr-neg 85.1%

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

      9. distribute-rgt-out 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in k around 0 99.6%

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

   if -1.55e28 < m < 0.014200000000000001

   1. Initial program 94.4%

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

      1. associate-/l* 94.5%

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

      2. remove-double-neg 94.5%

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

      3. distribute-frac-neg2 94.5%

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

      4. distribute-neg-frac2 94.5%

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

      5. remove-double-neg 94.5%

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

      6. sqr-neg 94.5%

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

      7. associate-+l+ 94.5%

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

      8. sqr-neg 94.5%

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

      9. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in m around 0 94.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{+28} \lor \neg \left(m \leq 0.0142\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.95d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a + (a * (k * ((k * 100.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999996

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in m around 0 71.3%

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

   if 1.94999999999999996 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. remove-double-neg 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-neg-frac2 73.6%

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

      5. remove-double-neg 73.6%

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

      6. sqr-neg 73.6%

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

      7. associate-+l+ 73.6%

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

      8. sqr-neg 73.6%

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

      9. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 2.8%

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

      1. *-commutative 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in k around 0 36.1%

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

   10. Taylor expanded in a around 0 40.4%

       \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(100 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1)
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))
   (* a (+ 1.0 (* k (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	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 <= 2.1d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (1.0d0 + (k * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.1:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * (1.0 + (k * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.1)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (1.0 + (k * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in m around 0 71.3%

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

   if 2.10000000000000009 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. remove-double-neg 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-neg-frac2 73.6%

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

      5. remove-double-neg 73.6%

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

      6. sqr-neg 73.6%

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

      7. associate-+l+ 73.6%

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

      8. sqr-neg 73.6%

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

      9. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 40.4%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 40.4%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.26 \cdot 10^{+87} \lor \neg \left(k \leq 0.024\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -1.26e+87) (not (<= k 0.024))) (* 0.1 (/ a k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.26e+87) || !(k <= 0.024)) {
		tmp = 0.1 * (a / 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 <= (-1.26d+87)) .or. (.not. (k <= 0.024d0))) then
        tmp = 0.1d0 * (a / 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 <= -1.26e+87) || !(k <= 0.024)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -1.26e+87) or not (k <= 0.024):
		tmp = 0.1 * (a / k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -1.26e+87) || !(k <= 0.024))
		tmp = Float64(0.1 * Float64(a / k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -1.26e+87) || ~((k <= 0.024)))
		tmp = 0.1 * (a / k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -1.26e+87], N[Not[LessEqual[k, 0.024]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -1.26000000000000005e87 or 0.024 < k

   1. Initial program 77.2%

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

      1. associate-/l* 77.3%

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

      2. remove-double-neg 77.3%

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

      3. distribute-frac-neg2 77.3%

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

      4. distribute-neg-frac2 77.3%

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

      5. remove-double-neg 77.3%

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

      6. sqr-neg 77.3%

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

      7. associate-+l+ 77.3%

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

      8. sqr-neg 77.3%

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

      9. distribute-rgt-out 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in m around 0 57.8%

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

   6. Taylor expanded in k around 0 21.3%

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

      1. *-commutative 21.3%

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

   8. Simplified 21.3%

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

   9. Taylor expanded in k around inf 21.3%

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

   if -1.26000000000000005e87 < k < 0.024

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.6%

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

   6. Taylor expanded in k around 0 37.9%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.26 \cdot 10^{+87} \lor \neg \left(k \leq 0.024\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;k \leq 0.024:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -5.7e+88) (/ 0.1 (/ k a)) (if (<= k 0.024) a (* 0.1 (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -5.7e+88) {
		tmp = 0.1 / (k / a);
	} else if (k <= 0.024) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5.7d+88)) then
        tmp = 0.1d0 / (k / a)
    else if (k <= 0.024d0) then
        tmp = a
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -5.7e+88) {
		tmp = 0.1 / (k / a);
	} else if (k <= 0.024) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -5.7e+88:
		tmp = 0.1 / (k / a)
	elif k <= 0.024:
		tmp = a
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -5.7e+88)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (k <= 0.024)
		tmp = a;
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -5.7e+88)
		tmp = 0.1 / (k / a);
	elseif (k <= 0.024)
		tmp = a;
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -5.7e+88], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.024], a, N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -5.70000000000000021e88

   1. Initial program 72.7%

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

      1. associate-/l* 72.7%

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

      2. remove-double-neg 72.7%

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

      3. distribute-frac-neg2 72.7%

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

      4. distribute-neg-frac2 72.7%

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

      5. remove-double-neg 72.7%

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

      6. sqr-neg 72.7%

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

      7. associate-+l+ 72.7%

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

      8. sqr-neg 72.7%

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

      9. distribute-rgt-out 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in m around 0 49.9%

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

   6. Taylor expanded in k around 0 21.7%

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

      1. *-commutative 21.7%

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

   8. Simplified 21.7%

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

   9. Taylor expanded in k around inf 21.7%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   10. Step-by-step derivation

       1. clear-num 23.8%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 23.8%

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

   11. Applied egg-rr 23.8%

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

   if -5.70000000000000021e88 < k < 0.024

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.6%

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

   6. Taylor expanded in k around 0 37.9%

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

   if 0.024 < k

   1. Initial program 78.8%

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

      1. associate-/l* 78.9%

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

      2. remove-double-neg 78.9%

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

      3. distribute-frac-neg2 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. remove-double-neg 78.9%

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

      6. sqr-neg 78.9%

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

      7. associate-+l+ 78.9%

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

      8. sqr-neg 78.9%

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

      9. distribute-rgt-out 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in m around 0 60.6%

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

   6. Taylor expanded in k around 0 21.1%

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

      1. *-commutative 21.1%

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

   8. Simplified 21.1%

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

   9. Taylor expanded in k around inf 21.1%

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

Alternative 8: 54.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.95)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (* a (+ 1.0 (* k (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.95d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (1.0d0 + (k * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.95:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (1.0 + (k * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.95)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.95)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (1.0 + (k * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.95], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999996

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in m around 0 71.3%

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

   if 1.94999999999999996 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. remove-double-neg 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-neg-frac2 73.6%

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

      5. remove-double-neg 73.6%

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

      6. sqr-neg 73.6%

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

      7. associate-+l+ 73.6%

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

      8. sqr-neg 73.6%

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

      9. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 40.4%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 40.4%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.0) (* a (/ 1.0 (+ 1.0 (* k k)))) (* a (+ 1.0 (* k (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = a * (1.0 / (1.0 + (k * k)));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	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 <= 2.0d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * k)))
    else
        tmp = a * (1.0d0 + (k * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.0) {
		tmp = a * (1.0 / (1.0 + (k * k)));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.0:
		tmp = a * (1.0 / (1.0 + (k * k)))
	else:
		tmp = a * (1.0 + (k * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.0)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * k))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.0)
		tmp = a * (1.0 / (1.0 + (k * k)));
	else
		tmp = a * (1.0 + (k * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.0], N[(a * N[(1.0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in m around 0 71.3%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 2 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. remove-double-neg 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-neg-frac2 73.6%

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

      5. remove-double-neg 73.6%

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

      6. sqr-neg 73.6%

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

      7. associate-+l+ 73.6%

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

      8. sqr-neg 73.6%

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

      9. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 40.4%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 40.4%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1) (/ a (+ 1.0 (* k k))) (* a (+ 1.0 (* k (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	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 <= 2.1d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * (1.0d0 + (k * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.1:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (1.0 + (k * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.1)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (1.0 + (k * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in m around 0 71.3%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 2.10000000000000009 < m

   1. Initial program 73.6%

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

      1. associate-/l* 73.6%

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

      2. remove-double-neg 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-neg-frac2 73.6%

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

      5. remove-double-neg 73.6%

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

      6. sqr-neg 73.6%

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

      7. associate-+l+ 73.6%

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

      8. sqr-neg 73.6%

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

      9. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 40.4%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 40.4%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 40.4%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 8.5e+16) (/ a (+ 1.0 (* k k))) (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e+16) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8.5d+16) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e+16) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.5e+16:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.5e+16)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8.5e+16)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.5e+16], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.5e16

   1. Initial program 96.1%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 70.2%

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

   6. Taylor expanded in k around inf 69.7%

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

   if 8.5e16 < m

   1. Initial program 73.8%

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

      1. associate-/l* 73.8%

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

      2. remove-double-neg 73.8%

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

      3. distribute-frac-neg2 73.8%

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

      4. distribute-neg-frac2 73.8%

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

      5. remove-double-neg 73.8%

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

      6. sqr-neg 73.8%

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

      7. associate-+l+ 73.8%

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

      8. sqr-neg 73.8%

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

      9. distribute-rgt-out 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in m around 0 2.8%

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

   6. Taylor expanded in k around 0 10.0%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 8.6e+15) (/ a (+ 1.0 (* k 10.0))) (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.6e+15) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8.6d+15) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.6e+15) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.6e+15:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.6e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8.6e+15)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.6e+15], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.6e15

   1. Initial program 96.1%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 70.2%

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

   6. Taylor expanded in k around 0 42.8%

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

      1. *-commutative 42.8%

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

   8. Simplified 42.8%

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

   if 8.6e15 < m

   1. Initial program 73.8%

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

      1. associate-/l* 73.8%

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

      2. remove-double-neg 73.8%

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

      3. distribute-frac-neg2 73.8%

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

      4. distribute-neg-frac2 73.8%

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

      5. remove-double-neg 73.8%

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

      6. sqr-neg 73.8%

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

      7. associate-+l+ 73.8%

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

      8. sqr-neg 73.8%

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

      9. distribute-rgt-out 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in m around 0 2.8%

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

   6. Taylor expanded in k around 0 10.0%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.024:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.024) (* a (+ 1.0 (* k -10.0))) (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.024) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.024d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.024) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.024:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.024)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.024)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.024], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.024

   1. Initial program 94.5%

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

      1. associate-/l* 94.5%

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

      2. remove-double-neg 94.5%

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

      3. distribute-frac-neg2 94.5%

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

      4. distribute-neg-frac2 94.5%

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

      5. remove-double-neg 94.5%

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

      6. sqr-neg 94.5%

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

      7. associate-+l+ 94.5%

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

      8. sqr-neg 94.5%

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

      9. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in m around 0 40.9%

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

   6. Taylor expanded in k around 0 34.5%

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

   if 0.024 < k

   1. Initial program 78.8%

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

      1. associate-/l* 78.9%

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

      2. remove-double-neg 78.9%

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

      3. distribute-frac-neg2 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. remove-double-neg 78.9%

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

      6. sqr-neg 78.9%

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

      7. associate-+l+ 78.9%

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

      8. sqr-neg 78.9%

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

      9. distribute-rgt-out 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in m around 0 60.6%

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

   6. Taylor expanded in k around 0 21.1%

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

      1. *-commutative 21.1%

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

   8. Simplified 21.1%

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

   9. Taylor expanded in k around inf 21.1%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.024:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.6% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 88.8%

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

   1. associate-/l* 88.8%

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

   2. remove-double-neg 88.8%

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

   3. distribute-frac-neg2 88.8%

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

   4. distribute-neg-frac2 88.8%

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

   5. remove-double-neg 88.8%

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

   6. sqr-neg 88.8%

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

   7. associate-+l+ 88.8%

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

   8. sqr-neg 88.8%

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

   9. distribute-rgt-out 88.8%

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

3. Simplified 88.8%

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

5. Taylor expanded in m around 0 48.1%

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

6. Taylor expanded in k around 0 21.3%

   \[\leadsto \color{blue}{a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))