Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 99.0%

Time: 12.5s

Alternatives: 14

Speedup: 1.0×

• 23.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-10} \lor \neg \left(m \leq 0.00023\right):\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3e-10) (not (<= m 0.00023)))
   (/ a (pow k (- m)))
   (/ 1.0 (+ (* (/ k a) (+ k 10.0)) (/ 1.0 a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3e-10) || !(m <= 0.00023)) {
		tmp = a / pow(k, -m);
	} else {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3d-10)) .or. (.not. (m <= 0.00023d0))) then
        tmp = a / (k ** -m)
    else
        tmp = 1.0d0 / (((k / a) * (k + 10.0d0)) + (1.0d0 / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3e-10) || !(m <= 0.00023)) {
		tmp = a / Math.pow(k, -m);
	} else {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -3e-10) or not (m <= 0.00023):
		tmp = a / math.pow(k, -m)
	else:
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3e-10) || !(m <= 0.00023))
		tmp = Float64(a / (k ^ Float64(-m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(k / a) * Float64(k + 10.0)) + Float64(1.0 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -3e-10) || ~((m <= 0.00023)))
		tmp = a / (k ^ -m);
	else
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3e-10], N[Not[LessEqual[m, 0.00023]], $MachinePrecision]], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3e-10 or 2.3000000000000001e-4 < m

   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 90.5%

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

      7. fma-def 90.5%

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

      8. +-commutative 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in k around 0 100.0%

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

   6. Taylor expanded in k around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. log-rec 56.8%

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

      3. distribute-rgt-neg-in 56.8%

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

      4. remove-double-neg 56.8%

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

      5. *-rgt-identity 56.8%

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

      6. exp-neg 56.8%

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

      7. distribute-lft-neg-out 56.8%

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

      8. neg-mul-1 56.8%

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

      9. *-rgt-identity 56.8%

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

      10. neg-mul-1 56.8%

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

      11. distribute-lft-neg-in 56.8%

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

      12. *-commutative 56.8%

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

      13. 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 -3e-10 < m < 2.3000000000000001e-4

   1. Initial program 88.7%

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

      1. associate-*l/ 88.7%

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

      2. sqr-neg 88.7%

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

      3. associate-+l+ 88.7%

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

      4. sqr-neg 88.7%

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

      5. distribute-rgt-out 88.7%

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

   3. Simplified 88.7%

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

      1. +-commutative 88.7%

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

      2. +-commutative 88.7%

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

      3. fma-udef 88.7%

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

      4. associate-/r/ 88.7%

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

      5. clear-num 88.7%

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

      6. frac-2neg 88.7%

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

      7. metadata-eval 88.7%

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

   6. Applied egg-rr 88.7%

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

      1. associate-/l/ 88.7%

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

      2. distribute-neg-frac 88.7%

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

      3. neg-sub0 88.7%

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

      4. metadata-eval 88.7%

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

      5. fma-udef 88.7%

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

      6. +-commutative 88.7%

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

      7. associate--r+ 88.7%

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

      8. metadata-eval 88.7%

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

      9. metadata-eval 88.7%

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

   8. Simplified 88.7%

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

      1. div-sub 88.6%

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

      2. sub-neg 88.6%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. sub-neg 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in m around 0 88.6%

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

       1. +-commutative 88.6%

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

       2. +-commutative 88.6%

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

       3. associate-*l/ 98.6%

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

       4. +-commutative 98.6%

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

   15. Simplified 98.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-10} \lor \neg \left(m \leq 0.00023\right):\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2e-55)
   (/ 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 <= 2e-55) {
		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 <= 2d-55) 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 <= 2e-55) {
		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 <= 2e-55:
		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 <= 2e-55)
		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 <= 2e-55)
		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, 2e-55], 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 < 1.99999999999999999e-55

   1. Initial program 95.2%

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

      1. associate-/l* 95.3%

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

      2. sqr-neg 95.3%

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

      3. associate-+l+ 95.3%

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

      4. +-commutative 95.3%

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

      5. sqr-neg 95.3%

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

      6. distribute-rgt-out 95.9%

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

      7. fma-def 95.9%

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

      8. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in k around 0 100.0%

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

   6. Taylor expanded in k around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. log-rec 50.7%

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

      3. distribute-rgt-neg-in 50.7%

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

      4. remove-double-neg 50.7%

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

      5. *-rgt-identity 50.7%

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

      6. exp-neg 50.7%

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

      7. distribute-lft-neg-out 50.7%

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

      8. neg-mul-1 50.7%

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

      9. *-rgt-identity 50.7%

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

      10. neg-mul-1 50.7%

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

      11. distribute-lft-neg-in 50.7%

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

      12. *-commutative 50.7%

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

      13. 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 1.99999999999999999e-55 < k

   1. Initial program 81.7%

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

      1. associate-*l/ 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-+l+ 80.7%

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

      4. sqr-neg 80.7%

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

      5. distribute-rgt-out 80.7%

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

   3. Simplified 80.7%

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

      1. +-commutative 80.7%

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

      2. +-commutative 80.7%

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

      3. fma-udef 80.7%

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

      4. associate-/r/ 81.7%

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

      5. clear-num 81.7%

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

      6. frac-2neg 81.7%

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

      7. metadata-eval 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-/l/ 81.7%

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

      2. distribute-neg-frac 81.7%

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

      3. neg-sub0 81.7%

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

      4. metadata-eval 81.7%

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

      5. fma-udef 81.7%

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

      6. +-commutative 81.7%

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

      7. associate--r+ 81.7%

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

      8. metadata-eval 81.7%

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

      9. metadata-eval 81.7%

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

   8. Simplified 81.7%

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

      1. div-sub 81.7%

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

      2. sub-neg 81.7%

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

      3. times-frac 96.9%

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

   10. Applied egg-rr 96.9%

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

       1. sub-neg 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-10} \lor \neg \left(m \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3e-10) (not (<= m 3.7e-7)))
   (* a (pow k m))
   (/ 1.0 (+ (* (/ k a) (+ k 10.0)) (/ 1.0 a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3e-10) || !(m <= 3.7e-7)) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3d-10)) .or. (.not. (m <= 3.7d-7))) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / (((k / a) * (k + 10.0d0)) + (1.0d0 / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3e-10) || !(m <= 3.7e-7)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -3e-10) or not (m <= 3.7e-7):
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3e-10) || !(m <= 3.7e-7))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(k / a) * Float64(k + 10.0)) + Float64(1.0 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -3e-10) || ~((m <= 3.7e-7)))
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3e-10], N[Not[LessEqual[m, 3.7e-7]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3e-10 or 3.70000000000000004e-7 < m

   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/ 88.7%

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

      2. sqr-neg 88.7%

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

      3. associate-+l+ 88.7%

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

      4. sqr-neg 88.7%

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

      5. distribute-rgt-out 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -3e-10 < m < 3.70000000000000004e-7

   1. Initial program 88.7%

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

      1. associate-*l/ 88.7%

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

      2. sqr-neg 88.7%

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

      3. associate-+l+ 88.7%

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

      4. sqr-neg 88.7%

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

      5. distribute-rgt-out 88.7%

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

   3. Simplified 88.7%

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

      1. +-commutative 88.7%

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

      2. +-commutative 88.7%

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

      3. fma-udef 88.7%

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

      4. associate-/r/ 88.7%

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

      5. clear-num 88.7%

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

      6. frac-2neg 88.7%

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

      7. metadata-eval 88.7%

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

   6. Applied egg-rr 88.7%

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

      1. associate-/l/ 88.7%

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

      2. distribute-neg-frac 88.7%

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

      3. neg-sub0 88.7%

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

      4. metadata-eval 88.7%

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

      5. fma-udef 88.7%

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

      6. +-commutative 88.7%

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

      7. associate--r+ 88.7%

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

      8. metadata-eval 88.7%

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

      9. metadata-eval 88.7%

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

   8. Simplified 88.7%

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

      1. div-sub 88.6%

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

      2. sub-neg 88.6%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. sub-neg 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in m around 0 88.6%

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

       1. +-commutative 88.6%

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

       2. +-commutative 88.6%

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

       3. associate-*l/ 98.6%

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

       4. +-commutative 98.6%

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

   15. Simplified 98.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-10} \lor \neg \left(m \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq -8 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{a}{k}\right)\\ \mathbf{elif}\;m \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.4e+157)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (if (<= m -8e+14)
     (* (+ 1.0 (/ -10.0 k)) (* (/ 1.0 k) (/ a k)))
     (if (<= m 0.007)
       (/ 1.0 (+ (* (/ k a) (+ k 10.0)) (/ 1.0 a)))
       (* a (+ 1.0 (* k (+ -10.0 (* k 99.0)))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.4e+157) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= -8e+14) {
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	} else if (m <= 0.007) {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.4d+157)) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= (-8d+14)) then
        tmp = (1.0d0 + ((-10.0d0) / k)) * ((1.0d0 / k) * (a / k))
    else if (m <= 0.007d0) then
        tmp = 1.0d0 / (((k / a) * (k + 10.0d0)) + (1.0d0 / a))
    else
        tmp = a * (1.0d0 + (k * ((-10.0d0) + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.4e+157) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else if (m <= -8e+14) {
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	} else if (m <= 0.007) {
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.4e+157:
		tmp = a / (1.0 + (k * (k + 10.0)))
	elif m <= -8e+14:
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k))
	elif m <= 0.007:
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a))
	else:
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.4e+157)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= -8e+14)
		tmp = Float64(Float64(1.0 + Float64(-10.0 / k)) * Float64(Float64(1.0 / k) * Float64(a / k)));
	elseif (m <= 0.007)
		tmp = Float64(1.0 / Float64(Float64(Float64(k / a) * Float64(k + 10.0)) + Float64(1.0 / a)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(-10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.4e+157)
		tmp = a / (1.0 + (k * (k + 10.0)));
	elseif (m <= -8e+14)
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	elseif (m <= 0.007)
		tmp = 1.0 / (((k / a) * (k + 10.0)) + (1.0 / a));
	else
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.4e+157], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -8e+14], N[(N[(1.0 + N[(-10.0 / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.007], N[(1.0 / N[(N[(N[(k / a), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(-10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -3.39999999999999979e157

   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 47.6%

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

   if -3.39999999999999979e157 < m < -8e14

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. sqr-neg 97.4%

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

      3. associate-+l+ 97.4%

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

      4. +-commutative 97.4%

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

      5. sqr-neg 97.4%

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

      6. distribute-rgt-out 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

      4. remove-double-div 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in m around 0 29.4%

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

   8. Taylor expanded in k around inf 63.7%

      \[\leadsto \color{blue}{-10 \cdot \frac{a}{{k}^{3}} + \frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \color{blue}{\frac{-10 \cdot a}{{k}^{3}}} + \frac{a}{{k}^{2}} \]

      2. cube-mult 63.7%

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

      3. unpow2 63.7%

         \[\leadsto \frac{-10 \cdot a}{k \cdot \color{blue}{{k}^{2}}} + \frac{a}{{k}^{2}} \]

      4. times-frac 63.7%

         \[\leadsto \color{blue}{\frac{-10}{k} \cdot \frac{a}{{k}^{2}}} + \frac{a}{{k}^{2}} \]

      5. distribute-lft1-in 63.7%

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

   10. Simplified 63.7%

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

       1. *-un-lft-identity 63.7%

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

       2. unpow2 63.7%

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

       3. times-frac 61.2%

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

   12. Applied egg-rr 61.2%

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

   if -8e14 < m < 0.00700000000000000015

   1. Initial program 89.4%

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

      1. associate-*l/ 89.4%

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

      2. sqr-neg 89.4%

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

      3. associate-+l+ 89.4%

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

      4. sqr-neg 89.4%

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

      5. distribute-rgt-out 89.4%

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

   3. Simplified 89.4%

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

      1. +-commutative 89.4%

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

      2. +-commutative 89.4%

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

      3. fma-udef 89.4%

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

      4. associate-/r/ 89.4%

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

      5. clear-num 89.4%

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

      6. frac-2neg 89.4%

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

      7. metadata-eval 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. associate-/l/ 89.4%

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

      2. distribute-neg-frac 89.4%

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

      3. neg-sub0 89.4%

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

      4. metadata-eval 89.4%

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

      5. fma-udef 89.4%

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

      6. +-commutative 89.4%

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

      7. associate--r+ 89.4%

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

      8. metadata-eval 89.4%

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

      9. metadata-eval 89.4%

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

   8. Simplified 89.4%

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

      1. div-sub 89.3%

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

      2. sub-neg 89.3%

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

      3. times-frac 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. sub-neg 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in m around 0 85.3%

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

       1. +-commutative 85.3%

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

       2. +-commutative 85.3%

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

       3. associate-*l/ 93.6%

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

       4. +-commutative 93.6%

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

   15. Simplified 93.6%

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

   if 0.00700000000000000015 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-+l+ 80.7%

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

      4. +-commutative 80.7%

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

      5. sqr-neg 80.7%

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

      6. distribute-rgt-out 80.7%

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

      7. fma-def 80.7%

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

      8. +-commutative 80.7%

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

   3. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. associate-/r/ 80.7%

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

      3. clear-num 80.7%

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

      4. remove-double-div 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 3.3%

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

   8. Taylor expanded in k around 0 29.7%

      \[\leadsto \color{blue}{\left(1 + \left(-10 \cdot k + 99 \cdot {k}^{2}\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. *-commutative 29.7%

         \[\leadsto \left(1 + \left(\color{blue}{k \cdot -10} + 99 \cdot {k}^{2}\right)\right) \cdot a \]

      2. *-commutative 29.7%

         \[\leadsto \left(1 + \left(k \cdot -10 + \color{blue}{{k}^{2} \cdot 99}\right)\right) \cdot a \]

      3. unpow2 29.7%

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

      4. associate-*l* 29.7%

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

      5. distribute-lft-out 29.7%

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

   10. Simplified 29.7%

       \[\leadsto \color{blue}{\left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)} \cdot a \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq -8 \cdot 10^{+14}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{a}{k}\right)\\ \mathbf{elif}\;m \leq 0.007:\\ \;\;\;\;\frac{1}{\frac{k}{a} \cdot \left(k + 10\right) + \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{if}\;m \leq -4.8 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -1.82 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{a}{k}\right)\\ \mathbf{elif}\;m \leq 0.007:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (+ 1.0 (* k (+ k 10.0))))))
   (if (<= m -4.8e+157)
     t_0
     (if (<= m -1.82e+15)
       (* (+ 1.0 (/ -10.0 k)) (* (/ 1.0 k) (/ a k)))
       (if (<= m 0.007) t_0 (* a (+ 1.0 (* k (+ -10.0 (* k 99.0))))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * (k + 10.0)));
	double tmp;
	if (m <= -4.8e+157) {
		tmp = t_0;
	} else if (m <= -1.82e+15) {
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	} else if (m <= 0.007) {
		tmp = t_0;
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (1.0d0 + (k * (k + 10.0d0)))
    if (m <= (-4.8d+157)) then
        tmp = t_0
    else if (m <= (-1.82d+15)) then
        tmp = (1.0d0 + ((-10.0d0) / k)) * ((1.0d0 / k) * (a / k))
    else if (m <= 0.007d0) then
        tmp = t_0
    else
        tmp = a * (1.0d0 + (k * ((-10.0d0) + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * (k + 10.0)));
	double tmp;
	if (m <= -4.8e+157) {
		tmp = t_0;
	} else if (m <= -1.82e+15) {
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	} else if (m <= 0.007) {
		tmp = t_0;
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (1.0 + (k * (k + 10.0)))
	tmp = 0
	if m <= -4.8e+157:
		tmp = t_0
	elif m <= -1.82e+15:
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k))
	elif m <= 0.007:
		tmp = t_0
	else:
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))))
	tmp = 0.0
	if (m <= -4.8e+157)
		tmp = t_0;
	elseif (m <= -1.82e+15)
		tmp = Float64(Float64(1.0 + Float64(-10.0 / k)) * Float64(Float64(1.0 / k) * Float64(a / k)));
	elseif (m <= 0.007)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(-10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (1.0 + (k * (k + 10.0)));
	tmp = 0.0;
	if (m <= -4.8e+157)
		tmp = t_0;
	elseif (m <= -1.82e+15)
		tmp = (1.0 + (-10.0 / k)) * ((1.0 / k) * (a / k));
	elseif (m <= 0.007)
		tmp = t_0;
	else
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.8e+157], t$95$0, If[LessEqual[m, -1.82e+15], N[(N[(1.0 + N[(-10.0 / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.007], t$95$0, N[(a * N[(1.0 + N[(k * N[(-10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.7999999999999999e157 or -1.82e15 < m < 0.00700000000000000015

   1. Initial program 92.7%

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

      1. associate-*l/ 92.7%

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

      2. sqr-neg 92.7%

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

      3. associate-+l+ 92.7%

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

      4. sqr-neg 92.7%

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

      5. distribute-rgt-out 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in m around 0 73.7%

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

   if -4.7999999999999999e157 < m < -1.82e15

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. sqr-neg 97.4%

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

      3. associate-+l+ 97.4%

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

      4. +-commutative 97.4%

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

      5. sqr-neg 97.4%

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

      6. distribute-rgt-out 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

      4. remove-double-div 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in m around 0 29.4%

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

   8. Taylor expanded in k around inf 63.7%

      \[\leadsto \color{blue}{-10 \cdot \frac{a}{{k}^{3}} + \frac{a}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \color{blue}{\frac{-10 \cdot a}{{k}^{3}}} + \frac{a}{{k}^{2}} \]

      2. cube-mult 63.7%

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

      3. unpow2 63.7%

         \[\leadsto \frac{-10 \cdot a}{k \cdot \color{blue}{{k}^{2}}} + \frac{a}{{k}^{2}} \]

      4. times-frac 63.7%

         \[\leadsto \color{blue}{\frac{-10}{k} \cdot \frac{a}{{k}^{2}}} + \frac{a}{{k}^{2}} \]

      5. distribute-lft1-in 63.7%

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

   10. Simplified 63.7%

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

       1. *-un-lft-identity 63.7%

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

       2. unpow2 63.7%

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

       3. times-frac 61.2%

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

   12. Applied egg-rr 61.2%

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

   if 0.00700000000000000015 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-+l+ 80.7%

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

      4. +-commutative 80.7%

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

      5. sqr-neg 80.7%

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

      6. distribute-rgt-out 80.7%

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

      7. fma-def 80.7%

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

      8. +-commutative 80.7%

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

   3. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. associate-/r/ 80.7%

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

      3. clear-num 80.7%

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

      4. remove-double-div 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 3.3%

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

   8. Taylor expanded in k around 0 29.7%

      \[\leadsto \color{blue}{\left(1 + \left(-10 \cdot k + 99 \cdot {k}^{2}\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. *-commutative 29.7%

         \[\leadsto \left(1 + \left(\color{blue}{k \cdot -10} + 99 \cdot {k}^{2}\right)\right) \cdot a \]

      2. *-commutative 29.7%

         \[\leadsto \left(1 + \left(k \cdot -10 + \color{blue}{{k}^{2} \cdot 99}\right)\right) \cdot a \]

      3. unpow2 29.7%

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

      4. associate-*l* 29.7%

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

      5. distribute-lft-out 29.7%

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

   10. Simplified 29.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.8 \cdot 10^{+157}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq -1.82 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \frac{-10}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{a}{k}\right)\\ \mathbf{elif}\;m \leq 0.007:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e+26)
   (* (/ a k) 0.1)
   (if (<= m 0.39) (* a (+ 1.0 (* k -10.0))) (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.39) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.6d+26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 0.39d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.39) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.6e+26:
		tmp = (a / k) * 0.1
	elif m <= 0.39:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e+26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 0.39)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.6e+26)
		tmp = (a / k) * 0.1;
	elseif (m <= 0.39)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e+26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 0.39], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.60000000000000024e26

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. sqr-neg 98.7%

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

      3. associate-+l+ 98.7%

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

      4. sqr-neg 98.7%

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

      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 39.9%

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

   6. Taylor expanded in k around 0 22.7%

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

      1. *-commutative 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in k around inf 30.5%

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

   if -3.60000000000000024e26 < m < 0.39000000000000001

   1. Initial program 89.7%

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

      1. associate-/l* 89.8%

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

      2. sqr-neg 89.8%

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

      3. associate-+l+ 89.8%

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

      4. +-commutative 89.8%

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

      5. sqr-neg 89.8%

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

      6. distribute-rgt-out 89.8%

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

      7. fma-def 89.8%

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

      8. +-commutative 89.8%

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

   3. Simplified 89.8%

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

      1. clear-num 89.7%

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

      2. associate-/r/ 89.7%

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

      3. clear-num 89.7%

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

      4. remove-double-div 89.7%

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in m around 0 82.9%

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

   8. Taylor expanded in k around 0 41.3%

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

   if 0.39000000000000001 < m

   1. Initial program 80.5%

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

      1. associate-/l* 80.5%

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

      2. sqr-neg 80.5%

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

      3. associate-+l+ 80.5%

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

      4. +-commutative 80.5%

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

      5. sqr-neg 80.5%

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

      6. distribute-rgt-out 80.5%

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

      7. fma-def 80.5%

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

      8. +-commutative 80.5%

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

   3. Simplified 80.5%

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

      1. clear-num 80.5%

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

      2. associate-/r/ 80.5%

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

      3. clear-num 80.5%

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

      4. remove-double-div 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in m around 0 3.1%

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

   8. Taylor expanded in k around 0 7.7%

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

   9. Taylor expanded in k around inf 21.8%

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

       1. *-commutative 21.8%

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

   11. Simplified 21.8%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 62000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e+26)
   (* (/ a k) 0.1)
   (if (<= m 62000000000000.0) (/ a (+ 1.0 (* k 10.0))) (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 62000000000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.6d+26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 62000000000000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 62000000000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.6e+26:
		tmp = (a / k) * 0.1
	elif m <= 62000000000000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e+26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 62000000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.6e+26)
		tmp = (a / k) * 0.1;
	elseif (m <= 62000000000000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e+26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 62000000000000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.60000000000000024e26

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. sqr-neg 98.7%

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

      3. associate-+l+ 98.7%

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

      4. sqr-neg 98.7%

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

      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 39.9%

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

   6. Taylor expanded in k around 0 22.7%

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

      1. *-commutative 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in k around inf 30.5%

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

   if -3.60000000000000024e26 < m < 6.2e13

   1. Initial program 87.9%

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

      1. associate-*l/ 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l+ 87.9%

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

      4. sqr-neg 87.9%

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

      5. distribute-rgt-out 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in m around 0 81.3%

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

   6. Taylor expanded in k around 0 52.3%

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

      1. *-commutative 52.3%

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

   8. Simplified 52.3%

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

   if 6.2e13 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 22.3%

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

       1. *-commutative 22.3%

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

   11. Simplified 22.3%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 62000000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.007:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.007)
   (/ a (+ 1.0 (* k (+ k 10.0))))
   (* a (+ 1.0 (* k (+ -10.0 (* k 99.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.007) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.007d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (1.0d0 + (k * ((-10.0d0) + (k * 99.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.007) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.007:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.007)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * Float64(-10.0 + Float64(k * 99.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.007)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (1.0 + (k * (-10.0 + (k * 99.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.007], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * N[(-10.0 + N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.00700000000000000015

   1. Initial program 93.7%

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

      1. associate-*l/ 93.7%

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

      2. sqr-neg 93.7%

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

      3. associate-+l+ 93.7%

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

      4. sqr-neg 93.7%

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

      5. distribute-rgt-out 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in m around 0 63.9%

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

   if 0.00700000000000000015 < m

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-+l+ 80.7%

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

      4. +-commutative 80.7%

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

      5. sqr-neg 80.7%

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

      6. distribute-rgt-out 80.7%

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

      7. fma-def 80.7%

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

      8. +-commutative 80.7%

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

   3. Simplified 80.7%

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

      1. clear-num 80.7%

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

      2. associate-/r/ 80.7%

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

      3. clear-num 80.7%

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

      4. remove-double-div 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in m around 0 3.3%

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

   8. Taylor expanded in k around 0 29.7%

      \[\leadsto \color{blue}{\left(1 + \left(-10 \cdot k + 99 \cdot {k}^{2}\right)\right)} \cdot a \]
   9. Step-by-step derivation

      1. *-commutative 29.7%

         \[\leadsto \left(1 + \left(\color{blue}{k \cdot -10} + 99 \cdot {k}^{2}\right)\right) \cdot a \]

      2. *-commutative 29.7%

         \[\leadsto \left(1 + \left(k \cdot -10 + \color{blue}{{k}^{2} \cdot 99}\right)\right) \cdot a \]

      3. unpow2 29.7%

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

      4. associate-*l* 29.7%

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

      5. distribute-lft-out 29.7%

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

   10. Simplified 29.7%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.007:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(-10 + k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 780000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.2e+26)
   (* (/ a k) 0.1)
   (if (<= m 780000000.0) a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 780000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5.2d+26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 780000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.2e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 780000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.2e+26:
		tmp = (a / k) * 0.1
	elif m <= 780000000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.2e+26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 780000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.2e+26)
		tmp = (a / k) * 0.1;
	elseif (m <= 780000000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.2e+26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 780000000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.20000000000000004e26

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. sqr-neg 98.7%

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

      3. associate-+l+ 98.7%

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

      4. sqr-neg 98.7%

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

      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 39.9%

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

   6. Taylor expanded in k around 0 22.7%

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

      1. *-commutative 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in k around inf 30.5%

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

   if -5.20000000000000004e26 < m < 7.8e8

   1. Initial program 87.9%

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

      1. associate-*l/ 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l+ 87.9%

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

      4. sqr-neg 87.9%

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

      5. distribute-rgt-out 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in m around 0 81.3%

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

   6. Taylor expanded in k around 0 40.5%

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

   if 7.8e8 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 21.1%

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

       1. *-commutative 21.1%

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

   11. Simplified 21.1%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 780000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 6800000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e+26)
   (* (/ a k) 0.1)
   (if (<= m 6800000.0) a (* k (* a -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 6800000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -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 <= (-3.6d+26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 6800000.0d0) then
        tmp = a
    else
        tmp = k * (a * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 6800000.0) {
		tmp = a;
	} else {
		tmp = k * (a * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.6e+26:
		tmp = (a / k) * 0.1
	elif m <= 6800000.0:
		tmp = a
	else:
		tmp = k * (a * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e+26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 6800000.0)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.6e+26)
		tmp = (a / k) * 0.1;
	elseif (m <= 6800000.0)
		tmp = a;
	else
		tmp = k * (a * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e+26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 6800000.0], a, N[(k * N[(a * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.60000000000000024e26

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. sqr-neg 98.7%

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

      3. associate-+l+ 98.7%

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

      4. sqr-neg 98.7%

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

      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 39.9%

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

   6. Taylor expanded in k around 0 22.7%

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

      1. *-commutative 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in k around inf 30.5%

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

   if -3.60000000000000024e26 < m < 6.8e6

   1. Initial program 87.9%

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

      1. associate-*l/ 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l+ 87.9%

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

      4. sqr-neg 87.9%

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

      5. distribute-rgt-out 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in m around 0 81.3%

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

   6. Taylor expanded in k around 0 40.5%

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

   if 6.8e6 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 21.1%

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

       1. *-commutative 21.1%

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

       2. associate-*r* 22.3%

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

       3. *-commutative 22.3%

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

       4. associate-*l* 21.1%

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

   11. Simplified 21.1%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 6800000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 360000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.6e+26)
   (* (/ a k) 0.1)
   (if (<= m 360000000.0) a (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 360000000.0) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.6d+26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 360000000.0d0) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.6e+26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 360000000.0) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.6e+26:
		tmp = (a / k) * 0.1
	elif m <= 360000000.0:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.6e+26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 360000000.0)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.6e+26)
		tmp = (a / k) * 0.1;
	elseif (m <= 360000000.0)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.6e+26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 360000000.0], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.60000000000000024e26

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. sqr-neg 98.7%

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

      3. associate-+l+ 98.7%

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

      4. sqr-neg 98.7%

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

      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 39.9%

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

   6. Taylor expanded in k around 0 22.7%

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

      1. *-commutative 22.7%

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

   8. Simplified 22.7%

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

   9. Taylor expanded in k around inf 30.5%

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

   if -3.60000000000000024e26 < m < 3.6e8

   1. Initial program 87.9%

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

      1. associate-*l/ 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l+ 87.9%

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

      4. sqr-neg 87.9%

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

      5. distribute-rgt-out 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in m around 0 81.3%

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

   6. Taylor expanded in k around 0 40.5%

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

   if 3.6e8 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 22.3%

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

       1. *-commutative 22.3%

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

   11. Simplified 22.3%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 360000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 450000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 450000000.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 450000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 450000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 450000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 450000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 450000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 450000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 450000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.5e8

   1. Initial program 92.7%

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

      1. associate-*l/ 92.7%

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

      2. sqr-neg 92.7%

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

      3. associate-+l+ 92.7%

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

      4. sqr-neg 92.7%

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

      5. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in m around 0 63.0%

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

   if 4.5e8 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 22.3%

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

       1. *-commutative 22.3%

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

   11. Simplified 22.3%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 450000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6800000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 6800000.0) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6800000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 6800000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 6800000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 6800000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6800000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 6800000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6800000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8e6

   1. Initial program 92.7%

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

      1. associate-*l/ 92.7%

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

      2. sqr-neg 92.7%

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

      3. associate-+l+ 92.7%

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

      4. sqr-neg 92.7%

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

      5. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in m around 0 63.0%

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

   6. Taylor expanded in k around 0 24.4%

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

   if 6.8e6 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. sqr-neg 82.5%

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

      6. distribute-rgt-out 82.5%

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

      7. fma-def 82.5%

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

      8. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. associate-/r/ 82.5%

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

      3. clear-num 82.5%

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

      4. remove-double-div 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in m around 0 3.2%

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

   8. Taylor expanded in k around 0 7.9%

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

   9. Taylor expanded in k around inf 21.1%

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

       1. *-commutative 21.1%

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

   11. Simplified 21.1%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6800000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.4% 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 89.5%

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

   1. associate-*l/ 88.7%

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

   2. sqr-neg 88.7%

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

   3. associate-+l+ 88.7%

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

   4. sqr-neg 88.7%

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

   5. distribute-rgt-out 89.1%

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

3. Simplified 89.1%

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

5. Taylor expanded in m around 0 44.3%

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

6. Taylor expanded in k around 0 18.0%

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

7. Final simplification 18.0%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))