Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 99.6%

Time: 12.7s

Alternatives: 12

Speedup: 1.0×

• 22.2% 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: 90.2% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + k \cdot \left(10 \cdot t\_1 + \frac{k}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)) (t_1 (/ 1.0 t_0)))
   (if (<= k 2e-48) t_0 (/ 1.0 (+ t_1 (* k (+ (* 10.0 t_1) (/ k t_0))))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double t_1 = 1.0 / t_0;
	double tmp;
	if (k <= 2e-48) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (k ** m) * a
    t_1 = 1.0d0 / t_0
    if (k <= 2d-48) then
        tmp = t_0
    else
        tmp = 1.0d0 / (t_1 + (k * ((10.0d0 * t_1) + (k / t_0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double t_1 = 1.0 / t_0;
	double tmp;
	if (k <= 2e-48) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	t_1 = 1.0 / t_0
	tmp = 0
	if k <= 2e-48:
		tmp = t_0
	else:
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (k <= 2e-48)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(t_1 + Float64(k * Float64(Float64(10.0 * t_1) + Float64(k / t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (k <= 2e-48)
		tmp = t_0;
	else
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[k, 2e-48], t$95$0, N[(1.0 / N[(t$95$1 + N[(k * N[(N[(10.0 * t$95$1), $MachinePrecision] + N[(k / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e-48

   1. Initial program 94.2%

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

      1. associate-/l* 94.2%

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

      2. remove-double-neg 94.2%

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

      3. distribute-frac-neg2 94.2%

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

      4. distribute-neg-frac2 94.2%

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

      5. remove-double-neg 94.2%

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

      6. sqr-neg 94.2%

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

      7. associate-+l+ 94.2%

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

      8. sqr-neg 94.2%

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

      9. distribute-rgt-out 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 1.9999999999999999e-48 < k

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

      1. distribute-lft-in 82.7%

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

      2. associate-+l+ 82.7%

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

      3. associate-*r/ 82.7%

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

      4. clear-num 82.5%

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

      5. associate-+l+ 82.5%

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

      6. distribute-lft-in 82.5%

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

      7. +-commutative 82.5%

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

      8. fma-define 82.5%

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

      9. +-commutative 82.5%

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

      10. *-commutative 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in k around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-48}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{k}^{m} \cdot a} + k \cdot \left(10 \cdot \frac{1}{{k}^{m} \cdot a} + \frac{k}{{k}^{m} \cdot a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;\frac{t\_0}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 10^{+246}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= (/ t_0 (+ (* k k) (+ 1.0 (* k 10.0)))) 1e+246)
     (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 1e+246) {
		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 = (k ** m) * a
    if ((t_0 / ((k * k) + (1.0d0 + (k * 10.0d0)))) <= 1d+246) 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 = Math.pow(k, m) * a;
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 1e+246) {
		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 = math.pow(k, m) * a
	tmp = 0
	if (t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 1e+246:
		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((k ^ m) * a)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0)))) <= 1e+246)
		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 = (k ^ m) * a;
	tmp = 0.0;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 1e+246)
		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[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+246], 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))) < 1.00000000000000007e246

   1. Initial program 96.9%

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

      1. associate-/l* 96.9%

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

      2. remove-double-neg 96.9%

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

      3. distribute-frac-neg2 96.9%

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

      4. distribute-neg-frac2 96.9%

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

      5. remove-double-neg 96.9%

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

      6. sqr-neg 96.9%

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

      7. associate-+l+ 96.9%

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

      8. sqr-neg 96.9%

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

      9. distribute-rgt-out 96.9%

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

   3. Simplified 96.9%

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

   if 1.00000000000000007e246 < (/.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 64.3%

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

      1. associate-/l* 64.3%

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

      2. remove-double-neg 64.3%

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

      3. distribute-frac-neg2 64.3%

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

      4. distribute-neg-frac2 64.3%

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

      5. remove-double-neg 64.3%

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

      6. sqr-neg 64.3%

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

      7. associate-+l+ 64.3%

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

      8. sqr-neg 64.3%

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

      9. distribute-rgt-out 64.3%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in k around 0 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 10^{+246}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 0.0045:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.2e-8)
   (* a (/ (pow k m) (+ 1.0 (* k 10.0))))
   (if (<= m 0.0045)
     (/ 1.0 (+ (/ 1.0 a) (* k (/ (+ k 10.0) a))))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.2e-8) {
		tmp = a * (pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.0045) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	} else {
		tmp = pow(k, m) * 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 (m <= (-2.2d-8)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * 10.0d0)))
    else if (m <= 0.0045d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k + 10.0d0) / a)))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.2e-8) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.0045) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.2e-8:
		tmp = a * (math.pow(k, m) / (1.0 + (k * 10.0)))
	elif m <= 0.0045:
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.2e-8)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * 10.0))));
	elseif (m <= 0.0045)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(k + 10.0) / a))));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.2e-8)
		tmp = a * ((k ^ m) / (1.0 + (k * 10.0)));
	elseif (m <= 0.0045)
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.2e-8], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0045], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.1999999999999998e-8

   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 27.0%

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

   7. Simplified 100.0%

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

   if -2.1999999999999998e-8 < m < 0.00449999999999999966

   1. Initial program 92.4%

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

      1. associate-/l* 92.4%

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

      2. remove-double-neg 92.4%

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

      3. distribute-frac-neg2 92.4%

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

      4. distribute-neg-frac2 92.4%

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

      5. remove-double-neg 92.4%

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

      6. sqr-neg 92.4%

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

      7. associate-+l+ 92.4%

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

      8. sqr-neg 92.4%

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

      9. distribute-rgt-out 92.4%

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

   3. Simplified 92.4%

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

      1. distribute-lft-in 92.4%

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

      2. associate-+l+ 92.4%

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

      3. associate-*r/ 92.4%

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

      4. clear-num 92.0%

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

      5. associate-+l+ 92.0%

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

      6. distribute-lft-in 92.0%

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

      7. +-commutative 92.0%

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

      8. fma-define 92.0%

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

      9. +-commutative 92.0%

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

      10. *-commutative 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in k around 0 98.6%

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

      1. inv-pow 98.6%

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

      2. add-sqr-sqrt 53.5%

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

      3. unpow-prod-down 53.4%

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

   9. Applied egg-rr 53.4%

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

       1. pow-sqr 53.5%

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

       2. fma-undefine 53.5%

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

       3. distribute-rgt-in 53.5%

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

       4. associate-*l/ 53.5%

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

       5. associate-*r/ 53.5%

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

       6. associate-/r* 53.5%

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

       7. *-commutative 53.5%

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

       8. distribute-rgt-out 53.5%

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

       9. metadata-eval 53.5%

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

   11. Simplified 53.5%

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

   12. Taylor expanded in m around 0 89.4%

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

       1. associate-/l* 96.1%

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

       2. +-commutative 96.1%

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

   14. Simplified 96.1%

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

   if 0.00449999999999999966 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 0.0045:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00033 \lor \neg \left(m \leq 0.0038\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.00033) (not (<= m 0.0038)))
   (* (pow k m) a)
   (/ 1.0 (+ (/ 1.0 a) (* k (/ (+ k 10.0) a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00033) || !(m <= 0.0038)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / 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 ((m <= (-0.00033d0)) .or. (.not. (m <= 0.0038d0))) then
        tmp = (k ** m) * a
    else
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k + 10.0d0) / a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00033) || !(m <= 0.0038)) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.00033) or not (m <= 0.0038):
		tmp = math.pow(k, m) * a
	else:
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.00033) || !(m <= 0.0038))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(k + 10.0) / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.00033) || ~((m <= 0.0038)))
		tmp = (k ^ m) * a;
	else
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.00033], N[Not[LessEqual[m, 0.0038]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.3e-4 or 0.00379999999999999999 < m

   1. Initial program 88.4%

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

      1. associate-/l* 88.4%

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

      2. remove-double-neg 88.4%

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

      3. distribute-frac-neg2 88.4%

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

      4. distribute-neg-frac2 88.4%

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

      5. remove-double-neg 88.4%

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

      6. sqr-neg 88.4%

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

      7. associate-+l+ 88.4%

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

      8. sqr-neg 88.4%

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

      9. distribute-rgt-out 88.4%

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

   3. Simplified 88.4%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 -3.3e-4 < m < 0.00379999999999999999

   1. Initial program 92.4%

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

      1. associate-/l* 92.4%

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

      2. remove-double-neg 92.4%

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

      3. distribute-frac-neg2 92.4%

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

      4. distribute-neg-frac2 92.4%

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

      5. remove-double-neg 92.4%

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

      6. sqr-neg 92.4%

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

      7. associate-+l+ 92.4%

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

      8. sqr-neg 92.4%

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

      9. distribute-rgt-out 92.4%

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

   3. Simplified 92.4%

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

      1. distribute-lft-in 92.4%

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

      2. associate-+l+ 92.4%

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

      3. associate-*r/ 92.4%

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

      4. clear-num 92.0%

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

      5. associate-+l+ 92.0%

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

      6. distribute-lft-in 92.0%

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

      7. +-commutative 92.0%

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

      8. fma-define 92.0%

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

      9. +-commutative 92.0%

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

      10. *-commutative 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in k around 0 98.6%

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

      1. inv-pow 98.6%

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

      2. add-sqr-sqrt 53.5%

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

      3. unpow-prod-down 53.4%

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

   9. Applied egg-rr 53.4%

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

       1. pow-sqr 53.5%

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

       2. fma-undefine 53.5%

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

       3. distribute-rgt-in 53.5%

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

       4. associate-*l/ 53.5%

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

       5. associate-*r/ 53.5%

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

       6. associate-/r* 53.5%

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

       7. *-commutative 53.5%

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

       8. distribute-rgt-out 53.5%

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

       9. metadata-eval 53.5%

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

   11. Simplified 53.5%

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

   12. Taylor expanded in m around 0 89.4%

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

       1. associate-/l* 96.1%

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

       2. +-commutative 96.1%

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

   14. Simplified 96.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00033 \lor \neg \left(m \leq 0.0038\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.4:
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)))
	else:
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(k + 10.0) / a))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	else
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.4], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 96.2%

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

      1. associate-/l* 96.2%

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

      2. remove-double-neg 96.2%

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

      3. distribute-frac-neg2 96.2%

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

      4. distribute-neg-frac2 96.2%

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

      5. remove-double-neg 96.2%

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

      6. sqr-neg 96.2%

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

      7. associate-+l+ 96.2%

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

      8. sqr-neg 96.2%

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

      9. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

      1. distribute-lft-in 96.2%

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

      2. associate-+l+ 96.2%

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

      3. associate-*r/ 96.2%

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

      4. clear-num 96.0%

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

      5. associate-+l+ 96.0%

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

      6. distribute-lft-in 96.0%

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

      7. +-commutative 96.0%

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

      8. fma-define 96.0%

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

      9. +-commutative 96.0%

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

      10. *-commutative 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in k around 0 80.2%

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

      1. inv-pow 80.2%

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

      2. add-sqr-sqrt 45.7%

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

      3. unpow-prod-down 45.7%

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

   9. Applied egg-rr 45.7%

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

       1. pow-sqr 45.7%

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

       2. fma-undefine 45.7%

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

       3. distribute-rgt-in 45.7%

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

       4. associate-*l/ 45.7%

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

       5. associate-*r/ 45.7%

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

       6. associate-/r* 45.7%

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

       7. *-commutative 45.7%

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

       8. distribute-rgt-out 55.9%

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

       9. metadata-eval 55.9%

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

   11. Simplified 55.9%

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

   12. Taylor expanded in m around 0 68.9%

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

       1. associate-/l* 70.0%

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

       2. +-commutative 70.0%

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

   14. Simplified 70.0%

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

   if 2.39999999999999991 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 32.0%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

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

Alternative 6: 54.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	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 + 10.0)));
	else
		tmp = a * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 96.2%

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

      1. associate-/l* 96.2%

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

      2. remove-double-neg 96.2%

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

      3. distribute-frac-neg2 96.2%

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

      4. distribute-neg-frac2 96.2%

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

      5. remove-double-neg 96.2%

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

      6. sqr-neg 96.2%

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

      7. associate-+l+ 96.2%

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

      8. sqr-neg 96.2%

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

      9. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in m around 0 69.0%

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

   if 2.10000000000000009 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 32.0%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

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

Alternative 7: 38.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7e+16) (/ 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 <= 7e+16) {
		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 <= 7d+16) 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 <= 7e+16) {
		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 <= 7e+16:
		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 <= 7e+16)
		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 <= 7e+16)
		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, 7e+16], 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 2 regimes

2. if m < 7e16

   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* 96.3%

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

      2. remove-double-neg 96.3%

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

      3. distribute-frac-neg2 96.3%

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

      4. distribute-neg-frac2 96.3%

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

      5. remove-double-neg 96.3%

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

      6. sqr-neg 96.3%

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

      7. associate-+l+ 96.3%

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

      8. sqr-neg 96.3%

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

      9. distribute-rgt-out 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in m around 0 68.2%

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

   6. Taylor expanded in k around 0 41.2%

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

      1. *-commutative 41.2%

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

   8. Simplified 41.2%

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

   if 7e16 < m

   1. Initial program 77.0%

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

      1. associate-/l* 77.0%

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

      2. remove-double-neg 77.0%

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

      3. distribute-frac-neg2 77.0%

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

      4. distribute-neg-frac2 77.0%

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

      5. remove-double-neg 77.0%

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

      6. sqr-neg 77.0%

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

      7. associate-+l+ 77.0%

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

      8. sqr-neg 77.0%

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

      9. distribute-rgt-out 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 32.6%

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

   7. Taylor expanded in k around inf 32.6%

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

      1. *-commutative 32.6%

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

   9. Simplified 32.6%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.6% accurate, 8.1× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.4) (/ 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 <= 2.4) {
		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 <= 2.4d0) 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 <= 2.4) {
		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 <= 2.4:
		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 <= 2.4)
		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 <= 2.4)
		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, 2.4], 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 < 2.39999999999999991

   1. Initial program 96.2%

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

      1. associate-/l* 96.2%

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

      2. remove-double-neg 96.2%

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

      3. distribute-frac-neg2 96.2%

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

      4. distribute-neg-frac2 96.2%

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

      5. remove-double-neg 96.2%

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

      6. sqr-neg 96.2%

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

      7. associate-+l+ 96.2%

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

      8. sqr-neg 96.2%

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

      9. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in m around 0 69.0%

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

   if 2.39999999999999991 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 32.0%

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

   7. Taylor expanded in k around inf 32.0%

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

      1. *-commutative 32.0%

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

   9. Simplified 32.0%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.7% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 5.6e+36) (/ a (+ 1.0 (* k 10.0))) (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.6d+36) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.6e+36:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.6e+36)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.6e+36)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.6e+36], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.6000000000000001e36

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 66.7%

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

   6. Taylor expanded in k around 0 40.3%

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

      1. *-commutative 40.3%

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

   8. Simplified 40.3%

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

   if 5.6000000000000001e36 < m

   1. Initial program 77.1%

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

      1. associate-/l* 77.1%

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

      2. remove-double-neg 77.1%

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

      3. distribute-frac-neg2 77.1%

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

      4. distribute-neg-frac2 77.1%

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

      5. remove-double-neg 77.1%

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

      6. sqr-neg 77.1%

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

      7. associate-+l+ 77.1%

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

      8. sqr-neg 77.1%

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

      9. distribute-rgt-out 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 10.0%

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

      1. *-commutative 10.0%

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

   8. Simplified 10.0%

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

   9. Taylor expanded in k around inf 21.6%

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

       1. *-commutative 21.6%

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

       2. associate-*r* 21.6%

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

   11. Simplified 21.6%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 25.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 5.6e+36) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * 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 (m <= 5.6d+36) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.6e+36:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.6e+36)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.6e+36)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.6e+36], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.6000000000000001e36

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in k around 0 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in m around 0 24.7%

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

   if 5.6000000000000001e36 < m

   1. Initial program 77.1%

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

      1. associate-/l* 77.1%

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

      2. remove-double-neg 77.1%

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

      3. distribute-frac-neg2 77.1%

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

      4. distribute-neg-frac2 77.1%

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

      5. remove-double-neg 77.1%

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

      6. sqr-neg 77.1%

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

      7. associate-+l+ 77.1%

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

      8. sqr-neg 77.1%

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

      9. distribute-rgt-out 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 10.0%

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

      1. *-commutative 10.0%

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

   8. Simplified 10.0%

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

   9. Taylor expanded in k around inf 21.6%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 5.6e+36) a (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.6d+36) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+36) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.6e+36:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.6e+36)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.6e+36)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.6e+36], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.6000000000000001e36

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in k around 0 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in m around 0 24.7%

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

   if 5.6000000000000001e36 < m

   1. Initial program 77.1%

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

      1. associate-/l* 77.1%

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

      2. remove-double-neg 77.1%

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

      3. distribute-frac-neg2 77.1%

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

      4. distribute-neg-frac2 77.1%

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

      5. remove-double-neg 77.1%

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

      6. sqr-neg 77.1%

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

      7. associate-+l+ 77.1%

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

      8. sqr-neg 77.1%

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

      9. distribute-rgt-out 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 10.0%

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

      1. *-commutative 10.0%

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

   8. Simplified 10.0%

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

   9. Taylor expanded in k around inf 21.6%

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

       1. *-commutative 21.6%

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

       2. associate-*r* 21.6%

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

   11. Simplified 21.6%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+36}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 20.0% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.7%

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

   1. associate-/l* 89.7%

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

   2. remove-double-neg 89.7%

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

   3. distribute-frac-neg2 89.7%

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

   4. distribute-neg-frac2 89.7%

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

   5. remove-double-neg 89.7%

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

   6. sqr-neg 89.7%

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

   7. associate-+l+ 89.7%

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

   8. sqr-neg 89.7%

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

   9. distribute-rgt-out 89.7%

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

3. Simplified 89.7%

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

5. Taylor expanded in k around 0 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in m around 0 17.9%

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

9. Final simplification 17.9%

   \[\leadsto a \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))