Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 97.9%

Time: 10.8s

Alternatives: 12

Speedup: 1.0×

• 21.8% 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: 90.6% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (* k k) (+ 1.0 (* k 10.0)))) 2e+303)
     (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+303) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} 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 / ((k * k) + (1.0d0 + (k * 10.0d0)))) <= 2d+303) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    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 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+303) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+303:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	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(k * k) + Float64(1.0 + Float64(k * 10.0)))) <= 2e+303)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	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 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+303)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	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[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

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))) < 2e303

   1. Initial program 98.6%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. distribute-frac-neg2 98.6%

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

      4. distribute-neg-frac2 98.6%

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

      5. remove-double-neg 98.6%

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

      6. sqr-neg 98.6%

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

      7. associate-+l+ 98.6%

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

      8. sqr-neg 98.6%

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

      9. distribute-rgt-out 98.6%

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

   3. Simplified 98.6%

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

   if 2e303 < (/.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 68.8%

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

      1. associate-/l* 68.8%

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

      2. remove-double-neg 68.8%

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

      3. distribute-frac-neg2 68.8%

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

      4. distribute-neg-frac2 68.8%

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

      5. remove-double-neg 68.8%

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

      6. sqr-neg 68.8%

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

      7. associate-+l+ 68.8%

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

      8. sqr-neg 68.8%

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

      9. distribute-rgt-out 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (pow (/ (sqrt (* a (pow k m))) (hypot 1.0 k)) 2.0))

C

double code(double a, double k, double m) {
	return pow((sqrt((a * pow(k, m))) / hypot(1.0, k)), 2.0);
}

Java

public static double code(double a, double k, double m) {
	return Math.pow((Math.sqrt((a * Math.pow(k, m))) / Math.hypot(1.0, k)), 2.0);
}

Python

def code(a, k, m):
	return math.pow((math.sqrt((a * math.pow(k, m))) / math.hypot(1.0, k)), 2.0)

Julia

function code(a, k, m)
	return Float64(sqrt(Float64(a * (k ^ m))) / hypot(1.0, k)) ^ 2.0
end

MATLAB

function tmp = code(a, k, m)
	tmp = (sqrt((a * (k ^ m))) / hypot(1.0, k)) ^ 2.0;
end

Wolfram

code[a_, k_, m_] := N[Power[N[(N[Sqrt[N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 93.0%

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

   1. *-commutative 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in k around 0 91.6%

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

   1. add-sqr-sqrt 67.7%

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

   2. sqrt-div 63.4%

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

   3. hypot-1-def 63.4%

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

   4. sqrt-div 63.4%

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

   5. hypot-1-def 66.0%

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

7. Applied egg-rr 66.0%

   \[\leadsto \color{blue}{\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}} \]
8. Step-by-step derivation

   1. unpow2 66.0%

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

9. Simplified 66.0%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2}} \]
10. Add Preprocessing

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 2.95:\\ \;\;\;\;\frac{t\_0}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= m 2.95) (/ t_0 (+ 1.0 (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 2.95) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 2.95d0) then
        tmp = t_0 / (1.0d0 + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 2.95) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 2.95:
		tmp = t_0 / (1.0 + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 2.95)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 2.95)
		tmp = t_0 / (1.0 + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 2.95], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.9500000000000002

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in k around 0 96.2%

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

   if 2.9500000000000002 < m

   1. Initial program 82.8%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. distribute-frac-neg2 82.8%

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

      4. distribute-neg-frac2 82.8%

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

      5. remove-double-neg 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-+l+ 82.8%

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

      8. sqr-neg 82.8%

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

      9. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.95:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.4e-5)
   (* a (/ (pow k m) (+ 1.0 (* k 10.0))))
   (if (<= m 4.2e-9) (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0))))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.4e-5) {
		tmp = a * (pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 4.2e-9) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * pow(k, m);
	}
	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 <= (-2.4d-5)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * 10.0d0)))
    else if (m <= 4.2d-9) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.4e-5) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 4.2e-9) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.4e-5:
		tmp = a * (math.pow(k, m) / (1.0 + (k * 10.0)))
	elif m <= 4.2e-9:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.4e-5)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * 10.0))));
	elseif (m <= 4.2e-9)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.4e-5)
		tmp = a * ((k ^ m) / (1.0 + (k * 10.0)));
	elseif (m <= 4.2e-9)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.4e-5], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e-9], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.4000000000000001e-5

   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 k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.4000000000000001e-5 < m < 4.20000000000000039e-9

   1. Initial program 96.3%

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

      1. associate-/l* 96.4%

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

      2. remove-double-neg 96.4%

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

      3. distribute-frac-neg2 96.4%

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

      4. distribute-neg-frac2 96.4%

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

      5. remove-double-neg 96.4%

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

      6. sqr-neg 96.4%

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

      7. associate-+l+ 96.4%

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

      8. sqr-neg 96.4%

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

      9. distribute-rgt-out 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in m around 0 96.3%

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

   if 4.20000000000000039e-9 < m

   1. Initial program 82.8%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. distribute-frac-neg2 82.8%

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

      4. distribute-neg-frac2 82.8%

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

      5. remove-double-neg 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-+l+ 82.8%

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

      8. sqr-neg 82.8%

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

      9. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0152 \lor \neg \left(m \leq 0.015\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.0152) (not (<= m 0.015)))
   (* 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 <= -0.0152) || !(m <= 0.015)) {
		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 <= (-0.0152d0)) .or. (.not. (m <= 0.015d0))) 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 <= -0.0152) || !(m <= 0.015)) {
		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 <= -0.0152) or not (m <= 0.015):
		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 <= -0.0152) || !(m <= 0.015))
		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 <= -0.0152) || ~((m <= 0.015)))
		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, -0.0152], N[Not[LessEqual[m, 0.015]], $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 < -0.0152 or 0.014999999999999999 < m

   1. Initial program 91.6%

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

      1. associate-/l* 91.6%

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

      2. remove-double-neg 91.6%

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

      3. distribute-frac-neg2 91.6%

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

      4. distribute-neg-frac2 91.6%

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

      5. remove-double-neg 91.6%

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

      6. sqr-neg 91.6%

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

      7. associate-+l+ 91.6%

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

      8. sqr-neg 91.6%

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

      9. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in k around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -0.0152 < m < 0.014999999999999999

   1. Initial program 96.3%

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

      1. associate-/l* 96.4%

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

      2. remove-double-neg 96.4%

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

      3. distribute-frac-neg2 96.4%

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

      4. distribute-neg-frac2 96.4%

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

      5. remove-double-neg 96.4%

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

      6. sqr-neg 96.4%

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

      7. associate-+l+ 96.4%

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

      8. sqr-neg 96.4%

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

      9. distribute-rgt-out 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in m around 0 96.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0152 \lor \neg \left(m \leq 0.015\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \frac{a}{k}\\ \mathbf{if}\;m \leq -2.1 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (/ a k))))
   (if (<= m -2.1e-93)
     t_0
     (if (<= m 1.9e-49) a (if (<= m 1.35e+15) t_0 (* -10.0 (* a k)))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (a / k);
	double tmp;
	if (m <= -2.1e-93) {
		tmp = t_0;
	} else if (m <= 1.9e-49) {
		tmp = a;
	} else if (m <= 1.35e+15) {
		tmp = t_0;
	} else {
		tmp = -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) :: t_0
    real(8) :: tmp
    t_0 = k * (a / k)
    if (m <= (-2.1d-93)) then
        tmp = t_0
    else if (m <= 1.9d-49) then
        tmp = a
    else if (m <= 1.35d+15) then
        tmp = t_0
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (a / k);
	double tmp;
	if (m <= -2.1e-93) {
		tmp = t_0;
	} else if (m <= 1.9e-49) {
		tmp = a;
	} else if (m <= 1.35e+15) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (a / k)
	tmp = 0
	if m <= -2.1e-93:
		tmp = t_0
	elif m <= 1.9e-49:
		tmp = a
	elif m <= 1.35e+15:
		tmp = t_0
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(a / k))
	tmp = 0.0
	if (m <= -2.1e-93)
		tmp = t_0;
	elseif (m <= 1.9e-49)
		tmp = a;
	elseif (m <= 1.35e+15)
		tmp = t_0;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (a / k);
	tmp = 0.0;
	if (m <= -2.1e-93)
		tmp = t_0;
	elseif (m <= 1.9e-49)
		tmp = a;
	elseif (m <= 1.35e+15)
		tmp = t_0;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(a / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.1e-93], t$95$0, If[LessEqual[m, 1.9e-49], a, If[LessEqual[m, 1.35e+15], t$95$0, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.1000000000000001e-93 or 1.8999999999999999e-49 < m < 1.35e15

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. remove-double-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. remove-double-neg 99.1%

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

      6. sqr-neg 99.1%

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

      7. associate-+l+ 99.1%

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

      8. sqr-neg 99.1%

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

      9. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in m around 0 40.8%

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

   6. Taylor expanded in k around 0 8.4%

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

   7. Taylor expanded in k around inf 14.5%

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

   8. Taylor expanded in k around 0 25.8%

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

   if -2.1000000000000001e-93 < m < 1.8999999999999999e-49

   1. Initial program 95.5%

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

      1. associate-/l* 95.5%

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

      2. remove-double-neg 95.5%

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

      3. distribute-frac-neg2 95.5%

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

      4. distribute-neg-frac2 95.5%

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

      5. remove-double-neg 95.5%

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

      6. sqr-neg 95.5%

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

      7. associate-+l+ 95.5%

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

      8. sqr-neg 95.5%

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

      9. distribute-rgt-out 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in m around 0 95.5%

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

   6. Taylor expanded in k around 0 55.5%

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

   if 1.35e15 < m

   1. Initial program 83.7%

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

      1. associate-/l* 83.7%

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

      2. remove-double-neg 83.7%

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

      3. distribute-frac-neg2 83.7%

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

      4. distribute-neg-frac2 83.7%

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

      5. remove-double-neg 83.7%

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

      6. sqr-neg 83.7%

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

      7. associate-+l+ 83.7%

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

      8. sqr-neg 83.7%

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

      9. distribute-rgt-out 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 6.3%

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

   7. Taylor expanded in k around inf 17.6%

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

Alternative 7: 40.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;k \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 2.4:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -54.0)
   (* k (/ a k))
   (if (<= m 2.4) (/ a (+ 1.0 (* k 10.0))) (* a (+ 1.0 (* k (* k 99.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -54.0) {
		tmp = k * (a / k);
	} else if (m <= 2.4) {
		tmp = a / (1.0 + (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 <= (-54.0d0)) then
        tmp = k * (a / k)
    else if (m <= 2.4d0) then
        tmp = a / (1.0d0 + (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 <= -54.0) {
		tmp = k * (a / k);
	} else if (m <= 2.4) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (1.0 + (k * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -54.0:
		tmp = k * (a / k)
	elif m <= 2.4:
		tmp = a / (1.0 + (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 <= -54.0)
		tmp = Float64(k * Float64(a / k));
	elseif (m <= 2.4)
		tmp = Float64(a / Float64(1.0 + 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 <= -54.0)
		tmp = k * (a / k);
	elseif (m <= 2.4)
		tmp = a / (1.0 + (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, -54.0], N[(k * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -54

   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 31.9%

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

   6. Taylor expanded in k around 0 3.0%

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

   7. Taylor expanded in k around inf 12.3%

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

   8. Taylor expanded in k around 0 20.2%

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

   if -54 < m < 2.39999999999999991

   1. Initial program 96.4%

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

      1. associate-/l* 96.4%

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

      2. remove-double-neg 96.4%

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

      3. distribute-frac-neg2 96.4%

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

      4. distribute-neg-frac2 96.4%

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

      5. remove-double-neg 96.4%

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

      6. sqr-neg 96.4%

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

      7. associate-+l+ 96.4%

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

      8. sqr-neg 96.4%

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

      9. distribute-rgt-out 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in m around 0 95.1%

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

   6. Taylor expanded in k around 0 65.7%

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

      1. *-commutative 66.9%

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

   8. Simplified 65.7%

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

   if 2.39999999999999991 < m

   1. Initial program 82.8%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. distribute-frac-neg2 82.8%

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

      4. distribute-neg-frac2 82.8%

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

      5. remove-double-neg 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-+l+ 82.8%

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

      8. sqr-neg 82.8%

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

      9. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 29.2%

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

   7. Taylor expanded in k around inf 29.2%

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

      1. *-commutative 29.2%

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

   9. Simplified 29.2%

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

Alternative 8: 36.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -18000:\\ \;\;\;\;k \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -18000.0)
   (* k (/ a k))
   (if (<= m 1.35e+15) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -18000.0) {
		tmp = k * (a / k);
	} else if (m <= 1.35e+15) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -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 <= (-18000.0d0)) then
        tmp = k * (a / k)
    else if (m <= 1.35d+15) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-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 <= -18000.0) {
		tmp = k * (a / k);
	} else if (m <= 1.35e+15) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -18000.0:
		tmp = k * (a / k)
	elif m <= 1.35e+15:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -18000.0)
		tmp = Float64(k * Float64(a / k));
	elseif (m <= 1.35e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -18000.0)
		tmp = k * (a / k);
	elseif (m <= 1.35e+15)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -18000.0], N[(k * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.35e+15], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -18000

   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 31.9%

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

   6. Taylor expanded in k around 0 3.0%

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

   7. Taylor expanded in k around inf 12.3%

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

   8. Taylor expanded in k around 0 20.2%

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

   if -18000 < m < 1.35e15

   1. Initial program 95.2%

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

      1. associate-/l* 95.2%

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

      2. remove-double-neg 95.2%

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

      3. distribute-frac-neg2 95.2%

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

      4. distribute-neg-frac2 95.2%

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

      5. remove-double-neg 95.2%

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

      6. sqr-neg 95.2%

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

      7. associate-+l+ 95.2%

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

      8. sqr-neg 95.2%

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

      9. distribute-rgt-out 95.2%

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

   3. Simplified 95.2%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 0 64.9%

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

      1. *-commutative 67.4%

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

   8. Simplified 64.9%

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

   if 1.35e15 < m

   1. Initial program 83.7%

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

      1. associate-/l* 83.7%

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

      2. remove-double-neg 83.7%

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

      3. distribute-frac-neg2 83.7%

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

      4. distribute-neg-frac2 83.7%

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

      5. remove-double-neg 83.7%

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

      6. sqr-neg 83.7%

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

      7. associate-+l+ 83.7%

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

      8. sqr-neg 83.7%

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

      9. distribute-rgt-out 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 6.3%

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

   7. Taylor expanded in k around inf 17.6%

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

Alternative 9: 55.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.0)
   (* 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 <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		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 <= 2.0:
		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 <= 2.0)
		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 <= 2.0)
		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, 2.0], 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

   1. Initial program 98.3%

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

      1. associate-/l* 98.3%

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

      2. remove-double-neg 98.3%

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

      3. distribute-frac-neg2 98.3%

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

      4. distribute-neg-frac2 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sqr-neg 98.3%

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

      7. associate-+l+ 98.3%

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

      8. sqr-neg 98.3%

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

      9. distribute-rgt-out 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in m around 0 61.5%

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

   if 2 < m

   1. Initial program 82.8%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. distribute-frac-neg2 82.8%

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

      4. distribute-neg-frac2 82.8%

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

      5. remove-double-neg 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-+l+ 82.8%

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

      8. sqr-neg 82.8%

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

      9. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 29.2%

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

   7. Taylor expanded in k around inf 29.2%

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

      1. *-commutative 29.2%

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

   9. Simplified 29.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% 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 98.3%

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

      1. associate-/l* 98.3%

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

      2. remove-double-neg 98.3%

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

      3. distribute-frac-neg2 98.3%

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

      4. distribute-neg-frac2 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sqr-neg 98.3%

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

      7. associate-+l+ 98.3%

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

      8. sqr-neg 98.3%

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

      9. distribute-rgt-out 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in m around 0 61.4%

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

   if 1.94999999999999996 < m

   1. Initial program 82.8%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. distribute-frac-neg2 82.8%

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

      4. distribute-neg-frac2 82.8%

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

      5. remove-double-neg 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-+l+ 82.8%

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

      8. sqr-neg 82.8%

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

      9. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 29.2%

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

   7. Taylor expanded in k around inf 29.2%

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

      1. *-commutative 29.2%

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

   9. Simplified 29.2%

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

4. Final simplification 50.5%

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

Alternative 11: 25.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.35e+15) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35e+15) {
		tmp = a;
	} else {
		tmp = -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 <= 1.35d+15) then
        tmp = a
    else
        tmp = (-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 <= 1.35e+15) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.35e+15:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.35e+15)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.35e+15)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.35e+15], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35e15

   1. Initial program 97.7%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in m around 0 61.1%

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

   6. Taylor expanded in k around 0 26.3%

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

   if 1.35e15 < m

   1. Initial program 83.7%

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

      1. associate-/l* 83.7%

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

      2. remove-double-neg 83.7%

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

      3. distribute-frac-neg2 83.7%

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

      4. distribute-neg-frac2 83.7%

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

      5. remove-double-neg 83.7%

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

      6. sqr-neg 83.7%

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

      7. associate-+l+ 83.7%

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

      8. sqr-neg 83.7%

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

      9. distribute-rgt-out 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 6.3%

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

   7. Taylor expanded in k around inf 17.6%

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

Alternative 12: 20.1% 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 93.0%

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

   1. associate-/l* 93.0%

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

   2. remove-double-neg 93.0%

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

   3. distribute-frac-neg2 93.0%

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

   4. distribute-neg-frac2 93.0%

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

   5. remove-double-neg 93.0%

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

   6. sqr-neg 93.0%

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

   7. associate-+l+ 93.0%

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

   8. sqr-neg 93.0%

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

   9. distribute-rgt-out 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in m around 0 41.6%

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

6. Taylor expanded in k around 0 18.8%

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

Reproduce

herbie shell --seed 2024090 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))