Falkner and Boettcher, Appendix A

Percentage Accurate: 89.8% → 99.1%

Time: 13.3s

Alternatives: 14

Speedup: 1.0×

• 21.6% 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: 89.8% 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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-97}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\frac{1}{\frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.5e-97)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m 0.052)
     (/ 1.0 (+ (* (/ (/ k a) (pow k m)) (+ k 10.0)) (/ 1.0 a)))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-97) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.052) {
		tmp = 1.0 / ((((k / a) / pow(k, m)) * (k + 10.0)) + (1.0 / 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 <= (-1.5d-97)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else if (m <= 0.052d0) then
        tmp = 1.0d0 / ((((k / a) / (k ** m)) * (k + 10.0d0)) + (1.0d0 / 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 <= -1.5e-97) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.052) {
		tmp = 1.0 / ((((k / a) / Math.pow(k, m)) * (k + 10.0)) + (1.0 / a));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.5e-97:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	elif m <= 0.052:
		tmp = 1.0 / ((((k / a) / math.pow(k, m)) * (k + 10.0)) + (1.0 / a))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.5e-97)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	elseif (m <= 0.052)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k / a) / (k ^ m)) * Float64(k + 10.0)) + Float64(1.0 / a)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.5e-97)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	elseif (m <= 0.052)
		tmp = 1.0 / ((((k / a) / (k ^ m)) * (k + 10.0)) + (1.0 / a));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.5e-97], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.052], N[(1.0 / N[(N[(N[(N[(k / a), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.50000000000000012e-97

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -1.50000000000000012e-97 < m < 0.0519999999999999976

   1. Initial program 90.7%

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

      1. associate-/l* 90.6%

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

      2. remove-double-neg 90.6%

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

      3. distribute-frac-neg2 90.6%

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

      4. distribute-neg-frac2 90.6%

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

      5. remove-double-neg 90.6%

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

      6. sqr-neg 90.6%

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

      7. associate-+l+ 90.6%

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

      8. sqr-neg 90.6%

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

      9. distribute-rgt-out 90.6%

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

   3. Simplified 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. associate-+l+ 90.6%

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

      3. associate-*r/ 90.7%

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

      4. clear-num 90.5%

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

      5. associate-+l+ 90.5%

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

      6. distribute-lft-in 90.5%

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

      7. +-commutative 90.5%

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

      8. fma-define 90.5%

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

      9. +-commutative 90.5%

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

      10. *-commutative 90.5%

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

   6. Applied egg-rr 90.5%

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

   7. Taylor expanded in k around 0 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. un-div-inv 99.6%

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

      4. associate-/r* 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. fma-undefine 99.6%

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

       3. distribute-rgt-in 99.6%

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

       4. associate-*l/ 99.6%

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

       5. associate-*r/ 99.6%

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

       6. associate-/r* 99.6%

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

       7. *-commutative 99.6%

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

       8. distribute-lft-out 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in m around 0 99.4%

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

   if 0.0519999999999999976 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-97}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\frac{1}{\frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;k \leq 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + k \cdot \left(10 \cdot t\_1 + \frac{k}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)) (t_1 (/ 1.0 t_0)))
   (if (<= k 1e-68) 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 <= 1e-68) {
		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 <= 1d-68) 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 <= 1e-68) {
		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 <= 1e-68:
		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 <= 1e-68)
		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 <= 1e-68)
		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, 1e-68], 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 < 1.00000000000000007e-68

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in k around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   if 1.00000000000000007e-68 < k

   1. Initial program 82.0%

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

      1. associate-/l* 81.9%

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

      2. remove-double-neg 81.9%

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

      3. distribute-frac-neg2 81.9%

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

      4. distribute-neg-frac2 81.9%

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

      5. remove-double-neg 81.9%

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

      6. sqr-neg 81.9%

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

      7. associate-+l+ 81.9%

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

      8. sqr-neg 81.9%

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

      9. distribute-rgt-out 81.9%

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

   3. Simplified 81.9%

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

      1. distribute-lft-in 81.9%

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

      2. associate-+l+ 81.9%

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

      3. associate-*r/ 82.0%

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

      4. clear-num 81.9%

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

      5. associate-+l+ 81.9%

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

      6. distribute-lft-in 81.9%

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

      7. +-commutative 81.9%

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

      8. fma-define 81.9%

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

      9. +-commutative 81.9%

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

      10. *-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

   7. Taylor expanded in k around 0 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-68}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{k}^{m} \cdot a} + k \cdot \left(10 \cdot \frac{1}{{k}^{m} \cdot a} + \frac{k}{{k}^{m} \cdot a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{t\_0}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 0.26)
     (/ 1.0 (+ (/ 1.0 t_0) (* (/ (/ k a) (pow k m)) (+ k 10.0))))
     (pow (sqrt t_0) 2.0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 0.26) {
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / pow(k, m)) * (k + 10.0)));
	} else {
		tmp = pow(sqrt(t_0), 2.0);
	}
	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 (m <= 0.26d0) then
        tmp = 1.0d0 / ((1.0d0 / t_0) + (((k / a) / (k ** m)) * (k + 10.0d0)))
    else
        tmp = sqrt(t_0) ** 2.0d0
    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 (m <= 0.26) {
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / Math.pow(k, m)) * (k + 10.0)));
	} else {
		tmp = Math.pow(Math.sqrt(t_0), 2.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if m <= 0.26:
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / math.pow(k, m)) * (k + 10.0)))
	else:
		tmp = math.pow(math.sqrt(t_0), 2.0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 0.26)
		tmp = Float64(1.0 / Float64(Float64(1.0 / t_0) + Float64(Float64(Float64(k / a) / (k ^ m)) * Float64(k + 10.0))));
	else
		tmp = sqrt(t_0) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (m <= 0.26)
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / (k ^ m)) * (k + 10.0)));
	else
		tmp = sqrt(t_0) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, 0.26], N[(1.0 / N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(N[(N[(k / a), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

      1. distribute-lft-in 95.1%

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

      2. associate-+l+ 95.1%

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

      3. associate-*r/ 95.1%

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

      4. clear-num 95.0%

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

      5. associate-+l+ 95.0%

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

      6. distribute-lft-in 95.0%

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

      7. +-commutative 95.0%

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

      8. fma-define 95.0%

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

      9. +-commutative 95.0%

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

      10. *-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

   7. Taylor expanded in k around 0 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. fma-define 84.8%

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

      3. un-div-inv 84.8%

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

      4. associate-/r* 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. *-lft-identity 84.8%

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

       2. fma-undefine 84.8%

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

       3. distribute-rgt-in 81.9%

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

       4. associate-*l/ 81.9%

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

       5. associate-*r/ 81.9%

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

       6. associate-/r* 81.9%

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

       7. *-commutative 81.9%

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

       8. distribute-lft-out 96.3%

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

   11. Simplified 96.3%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 81.9%

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

      2. pow2 81.9%

         \[\leadsto \color{blue}{{\left(\sqrt{{k}^{m} \cdot a}\right)}^{2}} \]

      3. *-commutative 81.9%

         \[\leadsto {\left(\sqrt{\color{blue}{a \cdot {k}^{m}}}\right)}^{2} \]

   9. Applied egg-rr 81.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{{k}^{m} \cdot a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{{k}^{m} \cdot a}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 0.26)
     (/ 1.0 (+ (/ 1.0 t_0) (* (/ (/ k a) (pow k m)) (+ k 10.0))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 0.26) {
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / pow(k, m)) * (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 (m <= 0.26d0) then
        tmp = 1.0d0 / ((1.0d0 / t_0) + (((k / a) / (k ** m)) * (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 (m <= 0.26) {
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / Math.pow(k, m)) * (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 m <= 0.26:
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / math.pow(k, m)) * (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 (m <= 0.26)
		tmp = Float64(1.0 / Float64(Float64(1.0 / t_0) + Float64(Float64(Float64(k / a) / (k ^ m)) * 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 (m <= 0.26)
		tmp = 1.0 / ((1.0 / t_0) + (((k / a) / (k ^ m)) * (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[m, 0.26], N[(1.0 / N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(N[(N[(k / a), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

      1. distribute-lft-in 95.1%

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

      2. associate-+l+ 95.1%

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

      3. associate-*r/ 95.1%

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

      4. clear-num 95.0%

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

      5. associate-+l+ 95.0%

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

      6. distribute-lft-in 95.0%

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

      7. +-commutative 95.0%

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

      8. fma-define 95.0%

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

      9. +-commutative 95.0%

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

      10. *-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

   7. Taylor expanded in k around 0 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. fma-define 84.8%

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

      3. un-div-inv 84.8%

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

      4. associate-/r* 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. *-lft-identity 84.8%

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

       2. fma-undefine 84.8%

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

       3. distribute-rgt-in 81.9%

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

       4. associate-*l/ 81.9%

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

       5. associate-*r/ 81.9%

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

       6. associate-/r* 81.9%

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

       7. *-commutative 81.9%

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

       8. distribute-lft-out 96.3%

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

   11. Simplified 96.3%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{{k}^{m} \cdot a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-105}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.0054:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.1e-105)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m 0.0054)
     (/ 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 <= -1.1e-105) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.0054) {
		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 <= (-1.1d-105)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else if (m <= 0.0054d0) 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 <= -1.1e-105) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else if (m <= 0.0054) {
		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 <= -1.1e-105:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	elif m <= 0.0054:
		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 <= -1.1e-105)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	elseif (m <= 0.0054)
		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 <= -1.1e-105)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	elseif (m <= 0.0054)
		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, -1.1e-105], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0054], 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 < -1.10000000000000002e-105

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -1.10000000000000002e-105 < m < 0.0054000000000000003

   1. Initial program 90.7%

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

      1. associate-/l* 90.6%

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

      2. remove-double-neg 90.6%

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

      3. distribute-frac-neg2 90.6%

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

      4. distribute-neg-frac2 90.6%

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

      5. remove-double-neg 90.6%

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

      6. sqr-neg 90.6%

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

      7. associate-+l+ 90.6%

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

      8. sqr-neg 90.6%

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

      9. distribute-rgt-out 90.6%

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

   3. Simplified 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. associate-+l+ 90.6%

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

      3. associate-*r/ 90.7%

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

      4. clear-num 90.5%

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

      5. associate-+l+ 90.5%

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

      6. distribute-lft-in 90.5%

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

      7. +-commutative 90.5%

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

      8. fma-define 90.5%

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

      9. +-commutative 90.5%

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

      10. *-commutative 90.5%

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

   6. Applied egg-rr 90.5%

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

   7. Taylor expanded in k around 0 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. un-div-inv 99.6%

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

      4. associate-/r* 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. fma-undefine 99.6%

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

       3. distribute-rgt-in 99.6%

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

       4. associate-*l/ 99.6%

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

       5. associate-*r/ 99.6%

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

       6. associate-/r* 99.6%

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

       7. *-commutative 99.6%

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

       8. distribute-lft-out 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in m around 0 90.3%

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

       1. associate-/l* 98.3%

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

       2. +-commutative 98.3%

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

   14. Simplified 98.3%

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

   if 0.0054000000000000003 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{-105}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 0.0054:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 0.00155:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.35e-18)
   (* a (/ (pow k m) (+ 1.0 (* k 10.0))))
   (if (<= m 0.00155)
     (/ 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 <= -1.35e-18) {
		tmp = a * (pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.00155) {
		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 <= (-1.35d-18)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * 10.0d0)))
    else if (m <= 0.00155d0) 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 <= -1.35e-18) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.00155) {
		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 <= -1.35e-18:
		tmp = a * (math.pow(k, m) / (1.0 + (k * 10.0)))
	elif m <= 0.00155:
		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 <= -1.35e-18)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * 10.0))));
	elseif (m <= 0.00155)
		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 <= -1.35e-18)
		tmp = a * ((k ^ m) / (1.0 + (k * 10.0)));
	elseif (m <= 0.00155)
		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, -1.35e-18], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00155], 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 < -1.34999999999999994e-18

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.34999999999999994e-18 < m < 0.00154999999999999995

   1. Initial program 91.4%

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

      1. associate-/l* 91.4%

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

      2. remove-double-neg 91.4%

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

      3. distribute-frac-neg2 91.4%

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

      4. distribute-neg-frac2 91.4%

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

      5. remove-double-neg 91.4%

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

      6. sqr-neg 91.4%

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

      7. associate-+l+ 91.4%

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

      8. sqr-neg 91.4%

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

      9. distribute-rgt-out 91.4%

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

   3. Simplified 91.4%

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

      1. distribute-lft-in 91.4%

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

      2. associate-+l+ 91.4%

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

      3. associate-*r/ 91.4%

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

      4. clear-num 91.2%

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

      5. associate-+l+ 91.2%

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

      6. distribute-lft-in 91.2%

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

      7. +-commutative 91.2%

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

      8. fma-define 91.2%

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

      9. +-commutative 91.2%

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

      10. *-commutative 91.2%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in k around 0 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. un-div-inv 99.6%

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

      4. associate-/r* 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. fma-undefine 99.6%

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

       3. distribute-rgt-in 99.6%

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

       4. associate-*l/ 99.6%

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

       5. associate-*r/ 99.6%

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

       6. associate-/r* 99.6%

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

       7. *-commutative 99.6%

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

       8. distribute-lft-out 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in m around 0 91.1%

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

       1. associate-/l* 98.4%

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

       2. +-commutative 98.4%

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

   14. Simplified 98.4%

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

   if 0.00154999999999999995 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 100.0%

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

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.074 \lor \neg \left(m \leq 0.00165\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.074) (not (<= m 0.00165)))
   (* (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 <= -0.074) || !(m <= 0.00165)) {
		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 <= (-0.074d0)) .or. (.not. (m <= 0.00165d0))) 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 <= -0.074) || !(m <= 0.00165)) {
		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 <= -0.074) or not (m <= 0.00165):
		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 <= -0.074) || !(m <= 0.00165))
		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 <= -0.074) || ~((m <= 0.00165)))
		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, -0.074], N[Not[LessEqual[m, 0.00165]], $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 < -0.0739999999999999963 or 0.00165 < m

   1. Initial program 89.6%

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

      1. associate-/l* 89.6%

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

      2. remove-double-neg 89.6%

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

      3. distribute-frac-neg2 89.6%

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

      4. distribute-neg-frac2 89.6%

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

      5. remove-double-neg 89.6%

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

      6. sqr-neg 89.6%

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

      7. associate-+l+ 89.6%

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

      8. sqr-neg 89.6%

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

      9. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in k around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   if -0.0739999999999999963 < m < 0.00165

   1. Initial program 91.7%

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

      1. associate-/l* 91.6%

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

      2. remove-double-neg 91.6%

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

      3. distribute-frac-neg2 91.6%

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

      4. distribute-neg-frac2 91.6%

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

      5. remove-double-neg 91.6%

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

      6. sqr-neg 91.6%

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

      7. associate-+l+ 91.6%

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

      8. sqr-neg 91.6%

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

      9. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

      1. distribute-lft-in 91.6%

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

      2. associate-+l+ 91.6%

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

      3. associate-*r/ 91.7%

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

      4. clear-num 91.5%

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

      5. associate-+l+ 91.5%

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

      6. distribute-lft-in 91.5%

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

      7. +-commutative 91.5%

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

      8. fma-define 91.5%

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

      9. +-commutative 91.5%

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

      10. *-commutative 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in k around 0 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. un-div-inv 99.6%

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

      4. associate-/r* 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. fma-undefine 99.6%

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

       3. distribute-rgt-in 99.6%

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

       4. associate-*l/ 99.6%

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

       5. associate-*r/ 99.6%

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

       6. associate-/r* 99.6%

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

       7. *-commutative 99.6%

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

       8. distribute-lft-out 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in m around 0 91.0%

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

       1. associate-/l* 98.1%

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

       2. +-commutative 98.1%

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

   14. Simplified 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.074 \lor \neg \left(m \leq 0.00165\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;a - a \cdot \left(k \cdot \left(10 + k \cdot -99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1e+33)
   (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a))
   (if (<= m 0.26)
     (/ 1.0 (+ (/ 1.0 a) (* k (/ (+ k 10.0) a))))
     (- a (* a (* k (+ 10.0 (* k -99.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1e+33) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else if (m <= 0.26) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	} else {
		tmp = a - (a * (k * (10.0 + (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 <= (-1d+33)) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else if (m <= 0.26d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k * ((k + 10.0d0) / a)))
    else
        tmp = a - (a * (k * (10.0d0 + (k * (-99.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1e+33) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else if (m <= 0.26) {
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	} else {
		tmp = a - (a * (k * (10.0 + (k * -99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1e+33:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	elif m <= 0.26:
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)))
	else:
		tmp = a - (a * (k * (10.0 + (k * -99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1e+33)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	elseif (m <= 0.26)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k * Float64(Float64(k + 10.0) / a))));
	else
		tmp = Float64(a - Float64(a * Float64(k * Float64(10.0 + Float64(k * -99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1e+33)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	elseif (m <= 0.26)
		tmp = 1.0 / ((1.0 / a) + (k * ((k + 10.0) / a)));
	else
		tmp = a - (a * (k * (10.0 + (k * -99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1e+33], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.26], 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[(a * N[(k * N[(10.0 + N[(k * -99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.9999999999999995e32

   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. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. clear-num 100.0%

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

      5. associate-+l+ 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in m around 0 43.7%

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

   if -9.9999999999999995e32 < m < 0.26000000000000001

   1. Initial program 92.1%

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

      1. associate-/l* 92.1%

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

      2. remove-double-neg 92.1%

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

      3. distribute-frac-neg2 92.1%

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

      4. distribute-neg-frac2 92.1%

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

      5. remove-double-neg 92.1%

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

      6. sqr-neg 92.1%

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

      7. associate-+l+ 92.1%

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

      8. sqr-neg 92.1%

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

      9. distribute-rgt-out 92.1%

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

   3. Simplified 92.1%

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

      1. distribute-lft-in 92.1%

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

      2. associate-+l+ 92.1%

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

      3. associate-*r/ 92.1%

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

      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 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. fma-define 98.7%

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

      3. un-div-inv 98.7%

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

      4. associate-/r* 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. *-lft-identity 98.7%

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

       2. fma-undefine 98.7%

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

       3. distribute-rgt-in 97.8%

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

       4. associate-*l/ 97.8%

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

       5. associate-*r/ 97.8%

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

       6. associate-/r* 97.8%

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

       7. *-commutative 97.8%

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

       8. distribute-lft-out 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in m around 0 86.1%

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

       1. associate-/l* 92.8%

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

       2. +-commutative 92.8%

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

   14. Simplified 92.8%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

      1. distribute-lft-in 80.7%

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

      2. associate-+l+ 80.7%

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

      3. associate-*r/ 80.7%

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

      4. clear-num 80.7%

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

      5. associate-+l+ 80.7%

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

      6. distribute-lft-in 80.7%

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

      7. +-commutative 80.7%

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

      8. fma-define 80.7%

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

      9. +-commutative 80.7%

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

      10. *-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 2.7%

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

   8. Taylor expanded in k around 0 24.7%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 24.7%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. metadata-eval 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      3. *-commutative 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

      4. mul-1-neg 24.7%

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

      5. distribute-rgt1-in 24.7%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + a \cdot -10\right) \]

      6. metadata-eval 24.7%

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

      7. *-commutative 24.7%

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

   10. Simplified 24.7%

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

   11. Taylor expanded in a around 0 29.2%

       \[\leadsto a + \color{blue}{-1 \cdot \left(a \cdot \left(k \cdot \left(10 + -99 \cdot k\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 29.2%

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

       2. *-commutative 29.2%

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

   13. Simplified 29.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;m \leq 0.26:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;a - a \cdot \left(k \cdot \left(10 + k \cdot -99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a - a \cdot \left(k \cdot \left(10 + k \cdot -99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (- a (* a (* k (+ 10.0 (* k -99.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a - (a * (k * (10.0 + (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 <= 0.26d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a - (a * (k * (10.0d0 + (k * (-99.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a - (a * (k * (10.0 + (k * -99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.26:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a - (a * (k * (10.0 + (k * -99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.26)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a - Float64(a * Float64(k * Float64(10.0 + Float64(k * -99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.26)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a - (a * (k * (10.0 + (k * -99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.26], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a - N[(a * N[(k * N[(10.0 + N[(k * -99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in m around 0 70.2%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

      1. distribute-lft-in 80.7%

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

      2. associate-+l+ 80.7%

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

      3. associate-*r/ 80.7%

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

      4. clear-num 80.7%

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

      5. associate-+l+ 80.7%

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

      6. distribute-lft-in 80.7%

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

      7. +-commutative 80.7%

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

      8. fma-define 80.7%

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

      9. +-commutative 80.7%

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

      10. *-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 2.7%

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

   8. Taylor expanded in k around 0 24.7%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 24.7%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. metadata-eval 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      3. *-commutative 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

      4. mul-1-neg 24.7%

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

      5. distribute-rgt1-in 24.7%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + a \cdot -10\right) \]

      6. metadata-eval 24.7%

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

      7. *-commutative 24.7%

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

   10. Simplified 24.7%

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

   11. Taylor expanded in a around 0 29.2%

       \[\leadsto a + \color{blue}{-1 \cdot \left(a \cdot \left(k \cdot \left(10 + -99 \cdot k\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 29.2%

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

       2. *-commutative 29.2%

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

   13. Simplified 29.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a - a \cdot \left(k \cdot \left(10 + k \cdot -99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (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 <= 0.26d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (k * (a * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (k * (a * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.26:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (k * (a * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.26)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.26)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (k * (a * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.26], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in m around 0 70.2%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

      1. distribute-lft-in 80.7%

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

      2. associate-+l+ 80.7%

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

      3. associate-*r/ 80.7%

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

      4. clear-num 80.7%

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

      5. associate-+l+ 80.7%

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

      6. distribute-lft-in 80.7%

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

      7. +-commutative 80.7%

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

      8. fma-define 80.7%

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

      9. +-commutative 80.7%

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

      10. *-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 2.7%

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

   8. Taylor expanded in k around 0 24.7%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 24.7%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. metadata-eval 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      3. *-commutative 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

      4. mul-1-neg 24.7%

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

      5. distribute-rgt1-in 24.7%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + a \cdot -10\right) \]

      6. metadata-eval 24.7%

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

      7. *-commutative 24.7%

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

   10. Simplified 24.7%

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

   11. Taylor expanded in k around inf 24.7%

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

       1. *-commutative 24.7%

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

       2. associate-*l* 24.7%

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

   13. Simplified 24.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26) (/ a (+ 1.0 (* k 10.0))) (+ a (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (a * (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 <= 0.26d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a + (k * (a * (k * 99.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (k * (a * (k * 99.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.26:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a + (k * (a * (k * 99.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.26)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a + Float64(k * Float64(a * Float64(k * 99.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.26)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a + (k * (a * (k * 99.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.26], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in m around 0 70.2%

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

   6. Taylor expanded in k around 0 41.9%

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

      1. *-commutative 77.0%

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

   8. Simplified 41.9%

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

   if 0.26000000000000001 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. remove-double-neg 80.7%

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

      3. distribute-frac-neg2 80.7%

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

      4. distribute-neg-frac2 80.7%

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

      5. remove-double-neg 80.7%

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

      6. sqr-neg 80.7%

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

      7. associate-+l+ 80.7%

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

      8. sqr-neg 80.7%

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

      9. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

      1. distribute-lft-in 80.7%

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

      2. associate-+l+ 80.7%

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

      3. associate-*r/ 80.7%

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

      4. clear-num 80.7%

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

      5. associate-+l+ 80.7%

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

      6. distribute-lft-in 80.7%

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

      7. +-commutative 80.7%

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

      8. fma-define 80.7%

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

      9. +-commutative 80.7%

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

      10. *-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 2.7%

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

   8. Taylor expanded in k around 0 24.7%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 24.7%

         \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(-10\right) \cdot a\right)} \]

      2. metadata-eval 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a\right) \]

      3. *-commutative 24.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{a \cdot -10}\right) \]

      4. mul-1-neg 24.7%

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

      5. distribute-rgt1-in 24.7%

         \[\leadsto a + k \cdot \left(\left(-k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)}\right) + a \cdot -10\right) \]

      6. metadata-eval 24.7%

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

      7. *-commutative 24.7%

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

   10. Simplified 24.7%

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

   11. Taylor expanded in k around inf 24.7%

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

       1. *-commutative 24.7%

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

       2. associate-*l* 24.7%

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

   13. Simplified 24.7%

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

Alternative 12: 28.5% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1 + k \cdot 10} \end{array} \]

FPCore

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

C

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

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 / (1.0d0 + (k * 10.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l* 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-frac-neg2 90.4%

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

   4. distribute-neg-frac2 90.4%

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

   5. remove-double-neg 90.4%

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

   6. sqr-neg 90.4%

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

   7. associate-+l+ 90.4%

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

   8. sqr-neg 90.4%

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

   9. distribute-rgt-out 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in m around 0 48.3%

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

6. Taylor expanded in k around 0 29.1%

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

   1. *-commutative 75.8%

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

8. Simplified 29.1%

   \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]
9. Add Preprocessing

Alternative 13: 20.9% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ a + -10 \cdot \left(k \cdot a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (+ a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	return a + (-10.0 * (k * 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 + ((-10.0d0) * (k * a))
end function

Java

public static double code(double a, double k, double m) {
	return a + (-10.0 * (k * a));
}

Python

def code(a, k, m):
	return a + (-10.0 * (k * a))

Julia

function code(a, k, m)
	return Float64(a + Float64(-10.0 * Float64(k * a)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = a + (-10.0 * (k * a));
end

Wolfram

code[a_, k_, m_] := N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/l* 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-frac-neg2 90.4%

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

   4. distribute-neg-frac2 90.4%

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

   5. remove-double-neg 90.4%

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

   6. sqr-neg 90.4%

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

   7. associate-+l+ 90.4%

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

   8. sqr-neg 90.4%

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

   9. distribute-rgt-out 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in m around 0 48.3%

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

6. Taylor expanded in k around 0 21.9%

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

7. Final simplification 21.9%

   \[\leadsto a + -10 \cdot \left(k \cdot a\right) \]
8. Add Preprocessing

Alternative 14: 20.1% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.4%

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

   1. associate-/l* 90.4%

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

   2. remove-double-neg 90.4%

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

   3. distribute-frac-neg2 90.4%

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

   4. distribute-neg-frac2 90.4%

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

   5. remove-double-neg 90.4%

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

   6. sqr-neg 90.4%

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

   7. associate-+l+ 90.4%

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

   8. sqr-neg 90.4%

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

   9. distribute-rgt-out 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in m around 0 48.3%

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

6. Taylor expanded in k around 0 21.8%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))