Falkner and Boettcher, Appendix A

Percentage Accurate: 100.0% → 100.0%

Time: 39.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. a = -6.345523484422262e+49
  2. k = 4.1334891586105655e+249
  3. m = 6.759880953206116e+186

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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-*r/ 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-rgt-out 100.0%

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

   5. fma-def 100.0%

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

   6. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (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 + (k * 10.0d0)) + (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-+l+ 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-*r/ 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-rgt-out 100.0%

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

   5. fma-def 100.0%

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

   6. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in k around 0 47.4%

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

   1. exp-to-pow 47.4%

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

   2. exp-to-pow 47.4%

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

   3. exp-to-pow 47.5%

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

   4. exp-to-pow 47.4%

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

   5. associate-*r* 47.4%

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

   6. exp-to-pow 47.4%

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

   7. *-commutative 47.4%

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

   8. exp-to-pow 81.1%

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

   9. *-commutative 81.1%

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

   10. distribute-lft1-in 99.0%

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

   11. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-+l+ 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in k around inf 99.4%

   \[\leadsto \frac{a}{\frac{1 + \color{blue}{{k}^{2}}}{{k}^{m}}} \]
5. Step-by-step derivation

   1. unpow2 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{a}{{k}^{\left(-m\right)}} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (/ a (pow k (- m))))

C

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

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)
end function

Java

public static double code(double a, double k, double m) {
	return a / Math.pow(k, -m);
}

Python

def code(a, k, m):
	return a / math.pow(k, -m)

Julia

function code(a, k, m)
	return Float64(a / (k ^ Float64(-m)))
end

MATLAB

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

Wolfram

code[a_, k_, m_] := N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]

Derivation

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%

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

   2. associate-+l+ 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in k around 0 65.2%

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

   1. exp-to-pow 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in k around 0 65.2%

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

   1. rec-exp 65.2%

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

   2. distribute-rgt-neg-out 65.2%

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

   3. exp-to-pow 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a \cdot {k}^{m} \end{array} \]

FPCore

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

C

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

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)
end function

Java

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

Python

def code(a, k, m):
	return a * math.pow(k, m)

Julia

function code(a, k, m)
	return Float64(a * (k ^ m))
end

MATLAB

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

Wolfram

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

Derivation

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-*r/ 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-rgt-out 100.0%

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

   5. fma-def 100.0%

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

   6. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in k around 0 65.2%

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

   1. exp-to-pow 98.8%

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

   2. *-commutative 98.8%

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

6. Simplified 98.8%

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

7. Final simplification 98.8%

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

Alternative 8: 54.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.136:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.136) (/ a (* k k)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.136) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.136d0)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.136) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.136:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.136)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.136)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.136], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.13600000000000001

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.0%

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

   5. Taylor expanded in k around inf 63.6%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 63.6%

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

   7. Simplified 63.6%

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

   if -0.13600000000000001 < m

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 50.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.136:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9: 52.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.4e-5) (/ a (* k k)) (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.4e-5) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.4d-5)) then
        tmp = a / (k * k)
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.4e-5) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.4e-5:
		tmp = a / (k * k)
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.4e-5)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.4e-5)
		tmp = a / (k * k);
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.4e-5], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -4.3999999999999999e-5

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.3%

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

   5. Taylor expanded in k around inf 63.3%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 63.3%

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

   7. Simplified 63.3%

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

   if -4.3999999999999999e-5 < m

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 72.0%

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

      1. exp-to-pow 72.0%

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

      2. exp-to-pow 72.0%

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

      3. exp-to-pow 72.1%

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

      4. exp-to-pow 72.0%

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

      5. associate-*r* 72.0%

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

      6. exp-to-pow 72.0%

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

      7. *-commutative 72.0%

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

      8. exp-to-pow 99.1%

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

      9. *-commutative 99.1%

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

      10. distribute-lft1-in 99.1%

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

      11. *-commutative 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around 0 49.2%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 10: 51.9% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m -4.4e-5) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.4e-5) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.4d-5)) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.4e-5) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.4e-5:
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.4e-5)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.4e-5)
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.4e-5], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -4.3999999999999999e-5

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 4.3%

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

   5. Taylor expanded in k around inf 63.3%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 63.3%

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

   7. Simplified 63.3%

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

   if -4.3999999999999999e-5 < m

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 50.1%

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

   5. Taylor expanded in k around 0 48.9%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 33.1% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-rgt-out 100.0%

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

   5. fma-def 100.0%

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

   6. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 33.4%

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

5. Taylor expanded in k around 0 32.6%

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

6. Final simplification 32.6%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023278 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))