Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.6%

Time: 13.7s

Alternatives: 11

Speedup: 1.1×

• 23.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 2e+162) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 2e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 2d+162) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 2e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 2e+162:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 2e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 2e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+162], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 1.9999999999999999e162

   1. Initial program 98.6%

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

   if 1.9999999999999999e162 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{-17} \lor \neg \left(m \leq 4.8 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.2e-17) (not (<= m 4.8e-13)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.2e-17) || !(m <= 4.8e-13)) {
		tmp = a * pow(k, m);
	} 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 <= (-5.2d-17)) .or. (.not. (m <= 4.8d-13))) then
        tmp = a * (k ** m)
    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 <= -5.2e-17) || !(m <= 4.8e-13)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -5.2e-17) or not (m <= 4.8e-13):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5.2e-17) || !(m <= 4.8e-13))
		tmp = Float64(a * (k ^ m));
	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 <= -5.2e-17) || ~((m <= 4.8e-13)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5.2e-17], N[Not[LessEqual[m, 4.8e-13]], $MachinePrecision]], N[(a * N[Power[k, m], $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 < -5.20000000000000006e-17 or 4.7999999999999997e-13 < m

   1. Initial program 89.6%

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

   2. Taylor expanded in k around 0 100.0%

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

   if -5.20000000000000006e-17 < m < 4.7999999999999997e-13

   1. Initial program 96.6%

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

      1. *-commutative 96.6%

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

      2. add-sqr-sqrt 96.6%

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

      3. times-frac 96.6%

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

      4. associate-+l+ 96.6%

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

      5. +-commutative 96.6%

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

      6. distribute-rgt-out 96.6%

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

      7. fma-def 96.6%

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

      8. associate-+l+ 96.6%

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

      9. +-commutative 96.6%

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

      10. distribute-rgt-out 96.6%

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

      11. fma-def 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in m around 0 96.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{-17} \lor \neg \left(m \leq 4.8 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 58.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 66000000.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 66000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 66000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 66000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 66000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 66000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 66000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 66000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 6.6e7

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. add-sqr-sqrt 96.7%

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

      3. times-frac 96.8%

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

      4. associate-+l+ 96.8%

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

      5. +-commutative 96.8%

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

      6. distribute-rgt-out 96.8%

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

      7. fma-def 96.8%

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

      8. associate-+l+ 96.8%

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

      9. +-commutative 96.8%

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

      10. distribute-rgt-out 96.8%

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

      11. fma-def 96.8%

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

   3. Applied egg-rr 96.8%

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

   4. Taylor expanded in m around 0 93.6%

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

   if 6.6e7 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0 80.5%

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

   3. Taylor expanded in m around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in k around inf 18.0%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 4: 47.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-295} \lor \neg \left(k \leq 0.23\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 5.2e-295) (not (<= k 0.23)))
   (/ a (* k k))
   (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 5.2e-295) || !(k <= 0.23)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 5.2d-295) .or. (.not. (k <= 0.23d0))) then
        tmp = a / (k * k)
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 5.2e-295) || !(k <= 0.23)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 5.2e-295) or not (k <= 0.23):
		tmp = a / (k * k)
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 5.2e-295) || !(k <= 0.23))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 5.2e-295) || ~((k <= 0.23)))
		tmp = a / (k * k);
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 5.2e-295], N[Not[LessEqual[k, 0.23]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.1999999999999997e-295 or 0.23000000000000001 < k

   1. Initial program 88.1%

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

   2. Taylor expanded in m around 0 38.7%

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

   3. Taylor expanded in k around inf 41.5%

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

      1. unpow2 41.5%

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

   5. Simplified 41.5%

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

   if 5.1999999999999997e-295 < k < 0.23000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 56.0%

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

   3. Taylor expanded in k around 0 55.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-295} \lor \neg \left(k \leq 0.23\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 5: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-294}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.55e-294)
   (/ a (* k k))
   (if (<= k 0.075) (+ a (* -10.0 (* a k))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.55e-294) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a / (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 (k <= 2.55d-294) then
        tmp = a / (k * k)
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.55e-294) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.55e-294:
		tmp = a / (k * k)
	elif k <= 0.075:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.55e-294)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.55e-294)
		tmp = a / (k * k);
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.55e-294], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.55000000000000003e-294

   1. Initial program 89.6%

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

   2. Taylor expanded in m around 0 15.4%

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

   3. Taylor expanded in k around inf 22.1%

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

      1. unpow2 22.1%

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

   5. Simplified 22.1%

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

   if 2.55000000000000003e-294 < k < 0.0749999999999999972

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 56.6%

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

   3. Taylor expanded in k around 0 56.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 86.6%

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

   2. Taylor expanded in m around 0 66.3%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

      2. distribute-rgt-in 65.0%

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

      3. +-commutative 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-294}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6: 48.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 66000000.0) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 66000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 66000000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 66000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 66000000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 66000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 66000000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 66000000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 6.6e7

   1. Initial program 96.8%

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

   2. Taylor expanded in k around 0 70.2%

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

   3. Taylor expanded in m around 0 67.1%

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

   if 6.6e7 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0 80.5%

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

   3. Taylor expanded in m around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in k around inf 18.0%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 7: 57.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 165000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 165000000000.0) (/ a (+ 1.0 (* k k))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 165000000000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 165000000000.0d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 165000000000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 165000000000.0:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 165000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 165000000000.0)
		tmp = a / (1.0 + (k * k));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 165000000000.0], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 1.65e11

   1. Initial program 96.8%

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

   2. Taylor expanded in m around 0 93.6%

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

   3. Taylor expanded in k around 0 90.7%

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

   if 1.65e11 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0 80.5%

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

   3. Taylor expanded in m around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in k around inf 18.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 165000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 46.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-294} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 4.3e-294) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 4.3e-294) || !(k <= 1.0)) {
		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 ((k <= 4.3d-294) .or. (.not. (k <= 1.0d0))) 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 ((k <= 4.3e-294) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 4.3e-294) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 4.3e-294) || !(k <= 1.0))
		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 ((k <= 4.3e-294) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 4.3e-294], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 4.30000000000000019e-294 or 1 < k

   1. Initial program 88.1%

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

   2. Taylor expanded in m around 0 38.7%

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

   3. Taylor expanded in k around inf 41.5%

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

      1. unpow2 41.5%

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

   5. Simplified 41.5%

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

   if 4.30000000000000019e-294 < k < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. times-frac 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. fma-def 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-rgt-out 99.9%

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

      11. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in m around 0 56.0%

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

   5. Taylor expanded in k around 0 54.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-294} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 9: 30.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8.5e-61)
   (/ a (* k 10.0))
   (if (<= m 66000000.0) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.5e-61) {
		tmp = a / (k * 10.0);
	} else if (m <= 66000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-8.5d-61)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 66000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.5e-61) {
		tmp = a / (k * 10.0);
	} else if (m <= 66000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8.5e-61:
		tmp = a / (k * 10.0)
	elif m <= 66000000.0:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8.5e-61)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 66000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8.5e-61)
		tmp = a / (k * 10.0);
	elseif (m <= 66000000.0)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8.5e-61], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 66000000.0], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.50000000000000016e-61

   1. Initial program 98.9%

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

   2. Taylor expanded in m around 0 38.4%

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

   3. Taylor expanded in k around inf 40.3%

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

      1. unpow2 40.3%

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

      2. distribute-rgt-in 40.3%

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

      3. +-commutative 40.3%

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

   5. Simplified 40.3%

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

   6. Taylor expanded in k around 0 19.9%

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

      1. *-commutative 19.9%

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

   8. Simplified 19.9%

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

   if -8.50000000000000016e-61 < m < 6.6e7

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. add-sqr-sqrt 97.6%

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

      3. times-frac 97.7%

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

      4. associate-+l+ 97.7%

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

      5. +-commutative 97.7%

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

      6. distribute-rgt-out 97.7%

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

      7. fma-def 97.7%

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

      8. associate-+l+ 97.7%

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

      9. +-commutative 97.7%

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

      10. distribute-rgt-out 97.7%

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

      11. fma-def 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in m around 0 95.6%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 6.6e7 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0 80.5%

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

   3. Taylor expanded in m around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in k around inf 18.0%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 66000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 10: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 66000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 66000000.0) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 66000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 66000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 66000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 66000000.0:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 66000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 66000000.0)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 66000000.0], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.6e7

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. add-sqr-sqrt 98.3%

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

      3. times-frac 98.3%

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

      4. associate-+l+ 98.3%

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

      5. +-commutative 98.3%

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

      6. distribute-rgt-out 98.3%

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

      7. fma-def 98.3%

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

      8. associate-+l+ 98.3%

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

      9. +-commutative 98.3%

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

      10. distribute-rgt-out 98.3%

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

      11. fma-def 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in m around 0 65.4%

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

   5. Taylor expanded in k around 0 29.6%

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

   if 6.6e7 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0 80.5%

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

   3. Taylor expanded in m around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in k around inf 18.0%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 66000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 20.2% 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 91.9%

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

   1. *-commutative 91.9%

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

   2. add-sqr-sqrt 91.9%

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

   3. times-frac 91.5%

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

   4. associate-+l+ 91.5%

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

   5. +-commutative 91.5%

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

   6. distribute-rgt-out 91.5%

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

   7. fma-def 91.5%

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

   8. associate-+l+ 91.5%

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

   9. +-commutative 91.5%

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

   10. distribute-rgt-out 91.5%

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

   11. fma-def 91.5%

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

3. Applied egg-rr 91.5%

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

4. Taylor expanded in m around 0 44.2%

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

5. Taylor expanded in k around 0 20.8%

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

6. Final simplification 20.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))