Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 98.8%

Time: 13.9s

Alternatives: 14

Speedup: 1.0×

• 23.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-65}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8.5e-65)
   (* (pow k m) (/ a (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m 1.8e-7)
     (/ 1.0 (+ (/ 1.0 a) (/ k (/ a (+ k 10.0)))))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.5e-65) {
		tmp = pow(k, m) * (a / (1.0 + (k * (k + 10.0))));
	} else if (m <= 1.8e-7) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-8.5d-65)) then
        tmp = (k ** m) * (a / (1.0d0 + (k * (k + 10.0d0))))
    else if (m <= 1.8d-7) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k / (a / (k + 10.0d0))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.5e-65) {
		tmp = Math.pow(k, m) * (a / (1.0 + (k * (k + 10.0))));
	} else if (m <= 1.8e-7) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8.5e-65:
		tmp = math.pow(k, m) * (a / (1.0 + (k * (k + 10.0))))
	elif m <= 1.8e-7:
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8.5e-65)
		tmp = Float64((k ^ m) * Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	elseif (m <= 1.8e-7)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k / Float64(a / Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8.5e-65)
		tmp = (k ^ m) * (a / (1.0 + (k * (k + 10.0))));
	elseif (m <= 1.8e-7)
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8.5e-65], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.8e-7], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k / N[(a / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.5000000000000003e-65

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -8.5000000000000003e-65 < m < 1.79999999999999997e-7

   1. Initial program 91.0%

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

      1. associate-*l/ 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

      1. +-commutative 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-udef 91.0%

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

      4. associate-/r/ 91.0%

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

      5. clear-num 90.9%

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

      6. frac-2neg 90.9%

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

      7. metadata-eval 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. associate-/l/ 90.9%

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

      2. distribute-neg-frac 90.9%

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

      3. neg-sub0 90.9%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 90.9%

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

      5. fma-udef 90.9%

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

      6. +-commutative 90.9%

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

      7. associate--r+ 90.9%

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

      8. metadata-eval 90.9%

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

      9. metadata-eval 90.9%

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

   8. Simplified 90.9%

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

      1. div-sub 90.9%

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

      2. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. clear-num 99.4%

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

       2. frac-times 99.5%

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

       3. *-un-lft-identity 99.5%

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

   12. Applied egg-rr 99.5%

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

   13. Taylor expanded in m around 0 89.9%

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

       1. associate-/l* 98.6%

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

       2. +-commutative 98.6%

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

   15. Simplified 98.6%

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

   if 1.79999999999999997e-7 < m

   1. Initial program 79.3%

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

      1. associate-*l/ 73.6%

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

      2. sqr-neg 73.6%

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

      3. associate-+l+ 73.6%

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

      4. sqr-neg 73.6%

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

      5. distribute-rgt-out 73.6%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-65}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k + 10}{{k}^{m} \cdot \frac{a}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.45e-18)
   (/ a (pow k (- m)))
   (/ -1.0 (- (/ -1.0 (* a (pow k m))) (/ (+ k 10.0) (* (pow k m) (/ a k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.45e-18) {
		tmp = a / pow(k, -m);
	} else {
		tmp = -1.0 / ((-1.0 / (a * pow(k, m))) - ((k + 10.0) / (pow(k, m) * (a / k))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.45d-18) then
        tmp = a / (k ** -m)
    else
        tmp = (-1.0d0) / (((-1.0d0) / (a * (k ** m))) - ((k + 10.0d0) / ((k ** m) * (a / k))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.45e-18) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = -1.0 / ((-1.0 / (a * Math.pow(k, m))) - ((k + 10.0) / (Math.pow(k, m) * (a / k))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.45e-18:
		tmp = a / math.pow(k, -m)
	else:
		tmp = -1.0 / ((-1.0 / (a * math.pow(k, m))) - ((k + 10.0) / (math.pow(k, m) * (a / k))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.45e-18)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / Float64(a * (k ^ m))) - Float64(Float64(k + 10.0) / Float64((k ^ m) * Float64(a / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.45e-18)
		tmp = a / (k ^ -m);
	else
		tmp = -1.0 / ((-1.0 / (a * (k ^ m))) - ((k + 10.0) / ((k ^ m) * (a / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.45e-18], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(k + 10.0), $MachinePrecision] / N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-18

   1. Initial program 94.2%

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

      1. associate-/l* 94.2%

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

      2. sqr-neg 94.2%

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

      3. associate-+l+ 94.2%

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

      4. +-commutative 94.2%

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

      5. sqr-neg 94.2%

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

      6. distribute-rgt-out 94.2%

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

      7. fma-def 94.2%

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

      8. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in k around 0 100.0%

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

   6. Taylor expanded in k around inf 53.2%

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

      1. rec-exp 53.2%

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

      2. mul-1-neg 53.2%

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

      3. remove-double-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. log-rec 53.2%

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

      6. distribute-lft-neg-in 53.2%

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

      7. distribute-rgt-neg-out 53.2%

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

      8. exp-to-pow 100.0%

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

   8. Simplified 100.0%

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

   if 1.45e-18 < k

   1. Initial program 83.3%

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

      1. associate-*l/ 79.3%

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

      2. sqr-neg 79.3%

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

      3. associate-+l+ 79.3%

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

      4. sqr-neg 79.3%

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

      5. distribute-rgt-out 79.3%

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

   3. Simplified 79.3%

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

      1. +-commutative 79.3%

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

      2. +-commutative 79.3%

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

      3. fma-udef 79.3%

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

      4. associate-/r/ 83.3%

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

      5. clear-num 83.3%

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

      6. frac-2neg 83.3%

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

      7. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/l/ 83.3%

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

      2. distribute-neg-frac 83.3%

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

      3. neg-sub0 83.3%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 83.3%

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

      5. fma-udef 83.3%

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

      6. +-commutative 83.3%

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

      7. associate--r+ 83.3%

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

      8. metadata-eval 83.3%

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

      9. metadata-eval 83.3%

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

   8. Simplified 83.3%

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

      1. div-sub 83.3%

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

      2. times-frac 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. clear-num 96.6%

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

       2. frac-times 97.7%

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

       3. *-un-lft-identity 97.7%

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

   12. Applied egg-rr 97.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k + 10}{{k}^{m} \cdot \frac{a}{k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.45e-18)
   (/ a (pow k (- m)))
   (/ -1.0 (- (/ -1.0 (* a (pow k m))) (* (/ k a) (/ (+ k 10.0) (pow k m)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.45e-18) {
		tmp = a / pow(k, -m);
	} else {
		tmp = -1.0 / ((-1.0 / (a * pow(k, m))) - ((k / a) * ((k + 10.0) / pow(k, m))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.45d-18) then
        tmp = a / (k ** -m)
    else
        tmp = (-1.0d0) / (((-1.0d0) / (a * (k ** m))) - ((k / a) * ((k + 10.0d0) / (k ** m))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.45e-18) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = -1.0 / ((-1.0 / (a * Math.pow(k, m))) - ((k / a) * ((k + 10.0) / Math.pow(k, m))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.45e-18:
		tmp = a / math.pow(k, -m)
	else:
		tmp = -1.0 / ((-1.0 / (a * math.pow(k, m))) - ((k / a) * ((k + 10.0) / math.pow(k, m))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.45e-18)
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / Float64(a * (k ^ m))) - Float64(Float64(k / a) * Float64(Float64(k + 10.0) / (k ^ m)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.45e-18)
		tmp = a / (k ^ -m);
	else
		tmp = -1.0 / ((-1.0 / (a * (k ^ m))) - ((k / a) * ((k + 10.0) / (k ^ m))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.45e-18], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.0 / N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(k / a), $MachinePrecision] * N[(N[(k + 10.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-18

   1. Initial program 94.2%

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

      1. associate-/l* 94.2%

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

      2. sqr-neg 94.2%

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

      3. associate-+l+ 94.2%

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

      4. +-commutative 94.2%

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

      5. sqr-neg 94.2%

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

      6. distribute-rgt-out 94.2%

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

      7. fma-def 94.2%

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

      8. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in k around 0 100.0%

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

   6. Taylor expanded in k around inf 53.2%

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

      1. rec-exp 53.2%

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

      2. mul-1-neg 53.2%

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

      3. remove-double-neg 53.2%

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

      4. *-commutative 53.2%

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

      5. log-rec 53.2%

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

      6. distribute-lft-neg-in 53.2%

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

      7. distribute-rgt-neg-out 53.2%

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

      8. exp-to-pow 100.0%

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

   8. Simplified 100.0%

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

   if 1.45e-18 < k

   1. Initial program 83.3%

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

      1. associate-*l/ 79.3%

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

      2. sqr-neg 79.3%

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

      3. associate-+l+ 79.3%

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

      4. sqr-neg 79.3%

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

      5. distribute-rgt-out 79.3%

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

   3. Simplified 79.3%

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

      1. +-commutative 79.3%

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

      2. +-commutative 79.3%

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

      3. fma-udef 79.3%

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

      4. associate-/r/ 83.3%

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

      5. clear-num 83.3%

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

      6. frac-2neg 83.3%

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

      7. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/l/ 83.3%

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

      2. distribute-neg-frac 83.3%

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

      3. neg-sub0 83.3%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 83.3%

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

      5. fma-udef 83.3%

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

      6. +-commutative 83.3%

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

      7. associate--r+ 83.3%

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

      8. metadata-eval 83.3%

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

      9. metadata-eval 83.3%

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

   8. Simplified 83.3%

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

      1. div-sub 83.3%

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

      2. times-frac 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -102 \lor \neg \left(m \leq 6.3 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -102.0) (not (<= m 6.3e-7)))
   (* a (pow k m))
   (/ 1.0 (+ (/ 1.0 a) (/ k (/ a (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -102.0) || !(m <= 6.3e-7)) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-102.0d0)) .or. (.not. (m <= 6.3d-7))) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / ((1.0d0 / a) + (k / (a / (k + 10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -102.0) || !(m <= 6.3e-7)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -102.0) or not (m <= 6.3e-7):
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -102.0) || !(m <= 6.3e-7))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k / Float64(a / Float64(k + 10.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -102.0) || ~((m <= 6.3e-7)))
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -102.0], N[Not[LessEqual[m, 6.3e-7]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k / N[(a / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -102 or 6.30000000000000003e-7 < m

   1. Initial program 88.9%

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

      1. associate-*l/ 85.8%

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

      2. sqr-neg 85.8%

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

      3. associate-+l+ 85.8%

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

      4. sqr-neg 85.8%

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

      5. distribute-rgt-out 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -102 < m < 6.30000000000000003e-7

   1. Initial program 91.8%

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

      1. associate-*l/ 91.8%

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

      2. sqr-neg 91.8%

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

      3. associate-+l+ 91.8%

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

      4. sqr-neg 91.8%

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

      5. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

      1. +-commutative 91.8%

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

      2. +-commutative 91.8%

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

      3. fma-udef 91.8%

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

      4. associate-/r/ 91.8%

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

      5. clear-num 91.6%

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

      6. frac-2neg 91.6%

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

      7. metadata-eval 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. associate-/l/ 91.6%

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

      2. distribute-neg-frac 91.6%

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

      3. neg-sub0 91.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 91.6%

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

      5. fma-udef 91.6%

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

      6. +-commutative 91.6%

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

      7. associate--r+ 91.6%

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

      8. metadata-eval 91.6%

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

      9. metadata-eval 91.6%

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

   8. Simplified 91.6%

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

      1. div-sub 91.6%

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

      2. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. clear-num 99.5%

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

       2. frac-times 99.5%

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

       3. *-un-lft-identity 99.5%

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

   12. Applied egg-rr 99.5%

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

   13. Taylor expanded in m around 0 90.0%

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

       1. associate-/l* 97.9%

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

       2. +-commutative 97.9%

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

   15. Simplified 97.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -102 \lor \neg \left(m \leq 6.3 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \left(k + 10\right) \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.8e-64)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (if (<= m 5200.0)
     (/ 1.0 (+ (/ 1.0 a) (* (+ k 10.0) (/ k a))))
     (- (* 10.0 (* k a)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.8e-64) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + ((k + 10.0) * (k / a)));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-2.8d-64)) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= 5200.0d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + ((k + 10.0d0) * (k / a)))
    else
        tmp = (10.0d0 * (k * a)) - a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.8e-64) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + ((k + 10.0) * (k / a)));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.8e-64:
		tmp = a / (1.0 + (k * (k + 10.0)))
	elif m <= 5200.0:
		tmp = 1.0 / ((1.0 / a) + ((k + 10.0) * (k / a)))
	else:
		tmp = (10.0 * (k * a)) - a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.8e-64)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= 5200.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(Float64(k + 10.0) * Float64(k / a))));
	else
		tmp = Float64(Float64(10.0 * Float64(k * a)) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.8e-64)
		tmp = a / (1.0 + (k * (k + 10.0)));
	elseif (m <= 5200.0)
		tmp = 1.0 / ((1.0 / a) + ((k + 10.0) * (k / a)));
	else
		tmp = (10.0 * (k * a)) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.8e-64], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5200.0], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(N[(k + 10.0), $MachinePrecision] * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.80000000000000004e-64

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 41.3%

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

   if -2.80000000000000004e-64 < m < 5200

   1. Initial program 91.2%

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

      1. associate-*l/ 91.2%

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

      2. sqr-neg 91.2%

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

      3. associate-+l+ 91.2%

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

      4. sqr-neg 91.2%

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

      5. distribute-rgt-out 91.2%

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

   3. Simplified 91.2%

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

      1. +-commutative 91.2%

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

      2. +-commutative 91.2%

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

      3. fma-udef 91.2%

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

      4. associate-/r/ 91.2%

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

      5. clear-num 91.1%

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

      6. frac-2neg 91.1%

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

      7. metadata-eval 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. associate-/l/ 91.1%

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

      2. distribute-neg-frac 91.1%

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

      3. neg-sub0 91.1%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 91.1%

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

      5. fma-udef 91.1%

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

      6. +-commutative 91.1%

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

      7. associate--r+ 91.1%

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

      8. metadata-eval 91.1%

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

      9. metadata-eval 91.1%

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

   8. Simplified 91.1%

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

      1. div-sub 91.1%

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

      2. times-frac 98.4%

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

   10. Applied egg-rr 98.4%

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

   11. Taylor expanded in m around 0 88.1%

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

       1. +-commutative 88.1%

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

       2. associate-*l/ 96.6%

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

       3. +-commutative 96.6%

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

   13. Simplified 96.6%

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

   if 5200 < m

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 13.4%

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

      5. sqr-neg 13.4%

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

      6. sqrt-unprod 1.2%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 12.0%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \left(k + 10\right) \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.4e-64)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (if (<= m 5200.0)
     (/ 1.0 (+ (/ 1.0 a) (/ k (/ a (+ k 10.0)))))
     (- (* 10.0 (* k a)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.4e-64) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-2.4d-64)) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= 5200.0d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k / (a / (k + 10.0d0))))
    else
        tmp = (10.0d0 * (k * a)) - a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.4e-64) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.4e-64:
		tmp = a / (1.0 + (k * (k + 10.0)))
	elif m <= 5200.0:
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))))
	else:
		tmp = (10.0 * (k * a)) - a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.4e-64)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= 5200.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k / Float64(a / Float64(k + 10.0)))));
	else
		tmp = Float64(Float64(10.0 * Float64(k * a)) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.4e-64)
		tmp = a / (1.0 + (k * (k + 10.0)));
	elseif (m <= 5200.0)
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	else
		tmp = (10.0 * (k * a)) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.4e-64], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5200.0], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k / N[(a / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.39999999999999998e-64

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 41.3%

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

   if -2.39999999999999998e-64 < m < 5200

   1. Initial program 91.2%

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

      1. associate-*l/ 91.2%

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

      2. sqr-neg 91.2%

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

      3. associate-+l+ 91.2%

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

      4. sqr-neg 91.2%

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

      5. distribute-rgt-out 91.2%

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

   3. Simplified 91.2%

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

      1. +-commutative 91.2%

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

      2. +-commutative 91.2%

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

      3. fma-udef 91.2%

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

      4. associate-/r/ 91.2%

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

      5. clear-num 91.1%

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

      6. frac-2neg 91.1%

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

      7. metadata-eval 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. associate-/l/ 91.1%

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

      2. distribute-neg-frac 91.1%

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

      3. neg-sub0 91.1%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 91.1%

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

      5. fma-udef 91.1%

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

      6. +-commutative 91.1%

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

      7. associate--r+ 91.1%

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

      8. metadata-eval 91.1%

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

      9. metadata-eval 91.1%

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

   8. Simplified 91.1%

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

      1. div-sub 91.1%

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

      2. times-frac 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. clear-num 98.3%

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

       2. frac-times 98.4%

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

       3. *-un-lft-identity 98.4%

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

   12. Applied egg-rr 98.4%

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

   13. Taylor expanded in m around 0 88.1%

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

       1. associate-/l* 96.6%

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

       2. +-commutative 96.6%

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

   15. Simplified 96.6%

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

   if 5200 < m

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 13.4%

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

      5. sqr-neg 13.4%

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

      6. sqrt-unprod 1.2%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 12.0%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -15200000:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(-a\right) - a}{a \cdot \left(-a\right)}}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -15200000.0)
   (/ 1.0 (/ (- (* (* k (+ k 10.0)) (- a)) a) (* a (- a))))
   (if (<= m 5200.0)
     (/ 1.0 (+ (/ 1.0 a) (/ k (/ a (+ k 10.0)))))
     (- (* 10.0 (* k a)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -15200000.0) {
		tmp = 1.0 / ((((k * (k + 10.0)) * -a) - a) / (a * -a));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = (10.0 * (k * a)) - 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 <= (-15200000.0d0)) then
        tmp = 1.0d0 / ((((k * (k + 10.0d0)) * -a) - a) / (a * -a))
    else if (m <= 5200.0d0) then
        tmp = 1.0d0 / ((1.0d0 / a) + (k / (a / (k + 10.0d0))))
    else
        tmp = (10.0d0 * (k * a)) - a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -15200000.0) {
		tmp = 1.0 / ((((k * (k + 10.0)) * -a) - a) / (a * -a));
	} else if (m <= 5200.0) {
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -15200000.0:
		tmp = 1.0 / ((((k * (k + 10.0)) * -a) - a) / (a * -a))
	elif m <= 5200.0:
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))))
	else:
		tmp = (10.0 * (k * a)) - a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -15200000.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k * Float64(k + 10.0)) * Float64(-a)) - a) / Float64(a * Float64(-a))));
	elseif (m <= 5200.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a) + Float64(k / Float64(a / Float64(k + 10.0)))));
	else
		tmp = Float64(Float64(10.0 * Float64(k * a)) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -15200000.0)
		tmp = 1.0 / ((((k * (k + 10.0)) * -a) - a) / (a * -a));
	elseif (m <= 5200.0)
		tmp = 1.0 / ((1.0 / a) + (k / (a / (k + 10.0))));
	else
		tmp = (10.0 * (k * a)) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -15200000.0], N[(1.0 / N[(N[(N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] - a), $MachinePrecision] / N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5200.0], N[(1.0 / N[(N[(1.0 / a), $MachinePrecision] + N[(k / N[(a / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.52e7

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. associate-/r/ 100.0%

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

      5. clear-num 100.0%

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

      6. frac-2neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 100.0%

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

      5. fma-udef 100.0%

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

      6. +-commutative 100.0%

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

      7. associate--r+ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 90.5%

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

      2. times-frac 90.5%

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

   10. Applied egg-rr 90.5%

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

   11. Taylor expanded in m around 0 36.5%

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

       1. +-commutative 36.5%

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

       2. associate-*l/ 27.6%

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

       3. +-commutative 27.6%

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

   13. Simplified 27.6%

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

       1. +-commutative 27.6%

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

       2. associate-*l/ 36.5%

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

       3. frac-2neg 36.5%

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

       4. metadata-eval 36.5%

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

       5. frac-add 48.4%

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

       6. +-commutative 48.4%

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

   15. Applied egg-rr 48.4%

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

   if -1.52e7 < m < 5200

   1. Initial program 92.0%

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

      1. associate-*l/ 92.0%

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

      2. sqr-neg 92.0%

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

      3. associate-+l+ 92.0%

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

      4. sqr-neg 92.0%

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

      5. distribute-rgt-out 92.0%

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

   3. Simplified 92.0%

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

      1. +-commutative 92.0%

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

      2. +-commutative 92.0%

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

      3. fma-udef 92.0%

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

      4. associate-/r/ 92.0%

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

      5. clear-num 91.9%

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

      6. frac-2neg 91.9%

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

      7. metadata-eval 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. associate-/l/ 91.9%

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

      2. distribute-neg-frac 91.9%

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

      3. neg-sub0 91.9%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 91.9%

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

      5. fma-udef 91.9%

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

      6. +-commutative 91.9%

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

      7. associate--r+ 91.9%

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

      8. metadata-eval 91.9%

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

      9. metadata-eval 91.9%

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

   8. Simplified 91.9%

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

      1. div-sub 91.9%

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

      2. times-frac 98.5%

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

   10. Applied egg-rr 98.5%

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

       1. clear-num 98.4%

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

       2. frac-times 98.5%

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

       3. *-un-lft-identity 98.5%

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

   12. Applied egg-rr 98.5%

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

   13. Taylor expanded in m around 0 87.5%

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

       1. associate-/l* 95.1%

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

       2. +-commutative 95.1%

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

   15. Simplified 95.1%

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

   if 5200 < m

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 13.4%

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

      5. sqr-neg 13.4%

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

      6. sqrt-unprod 1.2%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 12.0%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -15200000:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot \left(k + 10\right)\right) \cdot \left(-a\right) - a}{a \cdot \left(-a\right)}}\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{\frac{a}{k + 10}}}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 9.6 \cdot 10^{+26} \lor \neg \left(m \leq 9.5 \cdot 10^{+250}\right):\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-1 + k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.32e-5)
   (* (/ a k) 0.1)
   (if (or (<= m 9.6e+26) (not (<= m 9.5e+250)))
     (* a (+ 1.0 (* k -10.0)))
     (* a (+ -1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.32e-5) {
		tmp = (a / k) * 0.1;
	} else if ((m <= 9.6e+26) || !(m <= 9.5e+250)) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * (-1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.32d-5)) then
        tmp = (a / k) * 0.1d0
    else if ((m <= 9.6d+26) .or. (.not. (m <= 9.5d+250))) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = a * ((-1.0d0) + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.32e-5) {
		tmp = (a / k) * 0.1;
	} else if ((m <= 9.6e+26) || !(m <= 9.5e+250)) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * (-1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.32e-5:
		tmp = (a / k) * 0.1
	elif (m <= 9.6e+26) or not (m <= 9.5e+250):
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = a * (-1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.32e-5)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif ((m <= 9.6e+26) || !(m <= 9.5e+250))
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(a * Float64(-1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.32e-5)
		tmp = (a / k) * 0.1;
	elseif ((m <= 9.6e+26) || ~((m <= 9.5e+250)))
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = a * (-1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.32e-5], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[Or[LessEqual[m, 9.6e+26], N[Not[LessEqual[m, 9.5e+250]], $MachinePrecision]], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.32000000000000007e-5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.6%

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

   6. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in k around inf 22.5%

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

   if -1.32000000000000007e-5 < m < 9.60000000000000018e26 or 9.49999999999999957e250 < 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. *-commutative 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in m around 0 67.9%

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

   6. Taylor expanded in k around 0 42.9%

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

      1. *-commutative 42.9%

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

   8. Simplified 42.9%

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

   9. Taylor expanded in k around 0 40.6%

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

       1. *-commutative 40.6%

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

   11. Simplified 40.6%

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

   12. Taylor expanded in a around 0 42.2%

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

       1. *-commutative 42.2%

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

   14. Simplified 42.2%

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

   if 9.60000000000000018e26 < m < 9.49999999999999957e250

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in m around 0 2.7%

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

      1. frac-2neg 2.7%

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

      2. unpow2 2.7%

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

      3. distribute-rgt-in 2.7%

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

      4. div-inv 2.7%

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

      5. distribute-neg-in 2.7%

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

      6. metadata-eval 2.7%

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

      7. +-commutative 2.7%

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

      8. sub-neg 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. un-div-inv 2.7%

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

      2. clear-num 2.7%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 12.5%

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

      5. sqr-neg 12.5%

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

      6. sqrt-unprod 1.1%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 15.4%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 9.6 \cdot 10^{+26} \lor \neg \left(m \leq 9.5 \cdot 10^{+250}\right):\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-1 + k \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-1 + k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.5e-5)
   (* (/ a k) 0.1)
   (if (<= m 5200.0) (/ a (+ 1.0 (* k 10.0))) (* a (+ -1.0 (* k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-5) {
		tmp = (a / k) * 0.1;
	} else if (m <= 5200.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (-1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.5d-5)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 5200.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * ((-1.0d0) + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-5) {
		tmp = (a / k) * 0.1;
	} else if (m <= 5200.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (-1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.5e-5:
		tmp = (a / k) * 0.1
	elif m <= 5200.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (-1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.5e-5)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 5200.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(-1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.5e-5)
		tmp = (a / k) * 0.1;
	elseif (m <= 5200.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (-1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.5e-5], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 5200.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.50000000000000004e-5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.6%

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

   6. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in k around inf 22.5%

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

   if -1.50000000000000004e-5 < m < 5200

   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. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in m around 0 88.9%

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

   6. Taylor expanded in k around 0 55.9%

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

      1. *-commutative 55.9%

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

   8. Simplified 55.9%

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

   if 5200 < m

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 13.4%

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

      5. sqr-neg 13.4%

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

      6. sqrt-unprod 1.2%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 12.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-1 + k \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.5e-5)
   (* (/ a k) 0.1)
   (if (<= m 5200.0) (/ a (+ 1.0 (* k 10.0))) (- (* 10.0 (* k a)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-5) {
		tmp = (a / k) * 0.1;
	} else if (m <= 5200.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (10.0 * (k * a)) - 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-5)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 5200.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (10.0d0 * (k * a)) - a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.5e-5) {
		tmp = (a / k) * 0.1;
	} else if (m <= 5200.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.5e-5:
		tmp = (a / k) * 0.1
	elif m <= 5200.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (10.0 * (k * a)) - a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.5e-5)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 5200.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(10.0 * Float64(k * a)) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.5e-5)
		tmp = (a / k) * 0.1;
	elseif (m <= 5200.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (10.0 * (k * a)) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.5e-5], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 5200.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.50000000000000004e-5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.6%

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

   6. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in k around inf 22.5%

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

   if -1.50000000000000004e-5 < m < 5200

   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. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in m around 0 88.9%

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

   6. Taylor expanded in k around 0 55.9%

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

      1. *-commutative 55.9%

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

   8. Simplified 55.9%

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

   if 5200 < m

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.4%

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

      4. sqrt-unprod 13.4%

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

      5. sqr-neg 13.4%

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

      6. sqrt-unprod 1.2%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 12.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 5200:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.6) (/ a (+ 1.0 (* k (+ k 10.0)))) (- (* 10.0 (* k a)) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.6) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (10.0 * (k * a)) - 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 <= 1.6d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (10.0d0 * (k * a)) - a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.6) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (10.0 * (k * a)) - a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.6:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (10.0 * (k * a)) - a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(10.0 * Float64(k * a)) - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = (10.0 * (k * a)) - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.6], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 95.5%

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

      1. associate-*l/ 95.4%

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

      2. sqr-neg 95.4%

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

      3. associate-+l+ 95.4%

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

      4. sqr-neg 95.4%

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

      5. distribute-rgt-out 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in m around 0 65.9%

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

   if 1.6000000000000001 < m

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in m around 0 2.8%

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

      1. frac-2neg 2.8%

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

      2. unpow2 2.8%

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

      3. distribute-rgt-in 2.8%

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

      4. div-inv 2.8%

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

      5. distribute-neg-in 2.8%

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

      6. metadata-eval 2.8%

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

      7. +-commutative 2.8%

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

      8. sub-neg 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. un-div-inv 2.8%

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

      2. clear-num 2.8%

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

      3. add-sqr-sqrt 1.3%

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

      4. sqrt-unprod 13.3%

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

      5. sqr-neg 13.3%

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

      6. sqrt-unprod 1.3%

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

      7. add-sqr-sqrt 2.4%

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

   9. Applied egg-rr 2.4%

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

       1. associate-/r/ 2.4%

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

       2. sub-neg 2.4%

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

       3. distribute-rgt-neg-in 2.4%

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

       4. +-commutative 2.4%

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

       5. distribute-neg-in 2.4%

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

       6. metadata-eval 2.4%

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

       7. unsub-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in k around 0 11.9%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;10 \cdot \left(k \cdot a\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.8e-8) (* (/ a k) 0.1) (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.8e-8) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5.8d-8)) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.8e-8) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.8e-8:
		tmp = (a / k) * 0.1
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.8e-8)
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.8e-8)
		tmp = (a / k) * 0.1;
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.8e-8], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.8000000000000003e-8

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.6%

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

   6. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in k around inf 22.5%

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

   if -5.8000000000000003e-8 < m

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in m around 0 47.8%

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

   6. Taylor expanded in k around 0 30.5%

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

      1. *-commutative 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in k around 0 29.9%

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

       1. *-commutative 29.9%

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

   11. Simplified 29.9%

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

   12. Taylor expanded in a around 0 31.0%

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

       1. *-commutative 31.0%

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

   14. Simplified 31.0%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 25.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m -1.05e-5) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e-5) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.05d-5)) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e-5) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.05e-5:
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e-5)
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.05e-5)
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e-5], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -1.04999999999999994e-5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.6%

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

   6. Taylor expanded in k around 0 13.2%

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

      1. *-commutative 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in k around inf 22.5%

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

   if -1.04999999999999994e-5 < m

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

      2. sqr-neg 85.5%

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

      3. associate-+l+ 85.5%

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

      4. +-commutative 85.5%

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

      5. sqr-neg 85.5%

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

      6. distribute-rgt-out 85.5%

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

      7. fma-def 85.5%

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

      8. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in k around 0 75.0%

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

   6. Taylor expanded in m around 0 28.0%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 19.9% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. sqr-neg 89.9%

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

   3. associate-+l+ 89.9%

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

   4. +-commutative 89.9%

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

   5. sqr-neg 89.9%

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

   6. distribute-rgt-out 89.9%

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

   7. fma-def 89.9%

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

   8. +-commutative 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in k around 0 81.9%

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

6. Taylor expanded in m around 0 20.6%

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

7. Final simplification 20.6%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))