Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.3%

Time: 10.3s

Alternatives: 12

Speedup: 1.1×

• 24.2% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (* k k) (+ 1.0 (* k 10.0)))) 2e+101)
     (/ t_0 (+ 1.0 (* k (+ k 10.0))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+101) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if ((t_0 / ((k * k) + (1.0d0 + (k * 10.0d0)))) <= 2d+101) then
        tmp = t_0 / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+101) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+101:
		tmp = t_0 / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0)))) <= 2e+101)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((k * k) + (1.0 + (k * 10.0)))) <= 2e+101)
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	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]}, If[LessEqual[N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+101], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 2e101

   1. Initial program 97.6%

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

      1. sqr-neg 97.6%

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

      2. associate-+l+ 97.6%

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

      3. sqr-neg 97.6%

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

      4. distribute-rgt-out 97.6%

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

   3. Simplified 97.6%

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

   if 2e101 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 63.3%

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

      1. sqr-neg 63.3%

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

      2. associate-+l+ 63.3%

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

      3. sqr-neg 63.3%

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

      4. distribute-rgt-out 63.3%

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

   3. Simplified 63.3%

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

   4. 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 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(1 + k \cdot 10\right)} \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-7} \lor \neg \left(m \leq 1.15 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -8.5e-7) (not (<= m 1.15e-6)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.5e-7) || !(m <= 1.15e-6)) {
		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 <= (-8.5d-7)) .or. (.not. (m <= 1.15d-6))) 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 <= -8.5e-7) || !(m <= 1.15e-6)) {
		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 <= -8.5e-7) or not (m <= 1.15e-6):
		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 <= -8.5e-7) || !(m <= 1.15e-6))
		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 <= -8.5e-7) || ~((m <= 1.15e-6)))
		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, -8.5e-7], N[Not[LessEqual[m, 1.15e-6]], $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 < -8.50000000000000014e-7 or 1.15e-6 < m

   1. Initial program 88.8%

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

      1. sqr-neg 88.8%

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

      2. associate-+l+ 88.8%

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

      3. sqr-neg 88.8%

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

      4. distribute-rgt-out 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -8.50000000000000014e-7 < m < 1.15e-6

   1. Initial program 94.9%

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

      1. sqr-neg 94.9%

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

      2. associate-+l+ 94.9%

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

      3. sqr-neg 94.9%

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

      4. distribute-rgt-out 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in m around 0 94.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-7} \lor \neg \left(m \leq 1.15 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 50.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\frac{1}{t_0 \cdot \frac{1}{a}}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(-a\right) \cdot \left(k \cdot 10\right) - a}{a \cdot \left(-a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.4)
     (/ 1.0 (* t_0 (/ 1.0 a)))
     (if (<= m 4.6e+21)
       (/ a (+ 1.0 t_0))
       (/ 1.0 (/ (- (* (- a) (* k 10.0)) a) (* a (- a))))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.4) {
		tmp = 1.0 / (t_0 * (1.0 / a));
	} else if (m <= 4.6e+21) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = 1.0 / (((-a * (k * 10.0)) - a) / (a * -a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-0.4d0)) then
        tmp = 1.0d0 / (t_0 * (1.0d0 / a))
    else if (m <= 4.6d+21) then
        tmp = a / (1.0d0 + t_0)
    else
        tmp = 1.0d0 / (((-a * (k * 10.0d0)) - a) / (a * -a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.4) {
		tmp = 1.0 / (t_0 * (1.0 / a));
	} else if (m <= 4.6e+21) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = 1.0 / (((-a * (k * 10.0)) - a) / (a * -a));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.4:
		tmp = 1.0 / (t_0 * (1.0 / a))
	elif m <= 4.6e+21:
		tmp = a / (1.0 + t_0)
	else:
		tmp = 1.0 / (((-a * (k * 10.0)) - a) / (a * -a))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.4)
		tmp = Float64(1.0 / Float64(t_0 * Float64(1.0 / a)));
	elseif (m <= 4.6e+21)
		tmp = Float64(a / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(-a) * Float64(k * 10.0)) - a) / Float64(a * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.4)
		tmp = 1.0 / (t_0 * (1.0 / a));
	elseif (m <= 4.6e+21)
		tmp = a / (1.0 + t_0);
	else
		tmp = 1.0 / (((-a * (k * 10.0)) - a) / (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.4], N[(1.0 / N[(t$95$0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.6e+21], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[((-a) * N[(k * 10.0), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.40000000000000002

   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. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 38.2%

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

      1. clear-num 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-udef 38.3%

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

      5. inv-pow 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   8. Simplified 38.3%

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

      1. div-inv 38.3%

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

   10. Applied egg-rr 38.3%

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

   11. Taylor expanded in k around inf 47.1%

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

       1. +-commutative 47.1%

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

       2. unpow2 47.1%

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

       3. distribute-rgt-in 47.1%

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

   13. Simplified 47.1%

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

   if -0.40000000000000002 < m < 4.6e21

   1. Initial program 94.1%

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

      1. sqr-neg 94.1%

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

      2. associate-+l+ 94.1%

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

      3. sqr-neg 94.1%

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

      4. distribute-rgt-out 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 90.8%

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

   if 4.6e21 < m

   1. Initial program 79.5%

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

      1. sqr-neg 79.5%

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

      2. associate-+l+ 79.5%

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

      3. sqr-neg 79.5%

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

      4. distribute-rgt-out 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 2.8%

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

      1. clear-num 2.8%

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

      2. +-commutative 2.8%

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

      3. +-commutative 2.8%

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

      4. fma-udef 2.8%

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

      5. inv-pow 2.8%

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

   6. Applied egg-rr 2.8%

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

      1. unpow-1 2.8%

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

   8. Simplified 2.8%

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

   9. Taylor expanded in k around 0 2.6%

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

       1. +-commutative 2.6%

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

       2. frac-2neg 2.6%

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

       3. metadata-eval 2.6%

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

       4. associate-*r/ 2.6%

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

       5. frac-add 11.5%

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

       6. neg-mul-1 11.5%

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

       7. *-commutative 11.5%

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

   11. Applied egg-rr 11.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\frac{1}{\left(k \cdot \left(k + 10\right)\right) \cdot \frac{1}{a}}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(-a\right) \cdot \left(k \cdot 10\right) - a}{a \cdot \left(-a\right)}}\\ \end{array} \]

Alternative 4: 46.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.28:\\ \;\;\;\;\frac{1}{t_0 \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.28) (/ 1.0 (* t_0 (/ 1.0 a))) (/ a (+ 1.0 t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.28) {
		tmp = 1.0 / (t_0 * (1.0 / a));
	} else {
		tmp = a / (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-0.28d0)) then
        tmp = 1.0d0 / (t_0 * (1.0d0 / a))
    else
        tmp = a / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.28) {
		tmp = 1.0 / (t_0 * (1.0 / a));
	} else {
		tmp = a / (1.0 + t_0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.28:
		tmp = 1.0 / (t_0 * (1.0 / a))
	else:
		tmp = a / (1.0 + t_0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.28)
		tmp = Float64(1.0 / Float64(t_0 * Float64(1.0 / a)));
	else
		tmp = Float64(a / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.28)
		tmp = 1.0 / (t_0 * (1.0 / a));
	else
		tmp = a / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.28], N[(1.0 / N[(t$95$0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -0.28000000000000003

   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. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 38.2%

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

      1. clear-num 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-udef 38.3%

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

      5. inv-pow 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   8. Simplified 38.3%

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

      1. div-inv 38.3%

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

   10. Applied egg-rr 38.3%

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

   11. Taylor expanded in k around inf 47.1%

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

       1. +-commutative 47.1%

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

       2. unpow2 47.1%

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

       3. distribute-rgt-in 47.1%

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

   13. Simplified 47.1%

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

   if -0.28000000000000003 < m

   1. Initial program 87.5%

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

      1. sqr-neg 87.5%

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

      2. associate-+l+ 87.5%

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

      3. sqr-neg 87.5%

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

      4. distribute-rgt-out 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in m around 0 50.9%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.28:\\ \;\;\;\;\frac{1}{\left(k \cdot \left(k + 10\right)\right) \cdot \frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5: 30.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.192:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot 10}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.192) (/ 1.0 (* 10.0 (/ k a))) (/ 1.0 (/ (+ 1.0 (* k 10.0)) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.192) {
		tmp = 1.0 / (10.0 * (k / a));
	} else {
		tmp = 1.0 / ((1.0 + (k * 10.0)) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.192d0)) then
        tmp = 1.0d0 / (10.0d0 * (k / a))
    else
        tmp = 1.0d0 / ((1.0d0 + (k * 10.0d0)) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.192) {
		tmp = 1.0 / (10.0 * (k / a));
	} else {
		tmp = 1.0 / ((1.0 + (k * 10.0)) / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.192:
		tmp = 1.0 / (10.0 * (k / a))
	else:
		tmp = 1.0 / ((1.0 + (k * 10.0)) / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.192)
		tmp = Float64(1.0 / Float64(10.0 * Float64(k / a)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * 10.0)) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.192)
		tmp = 1.0 / (10.0 * (k / a));
	else
		tmp = 1.0 / ((1.0 + (k * 10.0)) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.192], N[(1.0 / N[(10.0 * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.192

   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. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 38.2%

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

      1. clear-num 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-udef 38.3%

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

      5. inv-pow 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in k around 0 18.8%

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

   10. Taylor expanded in k around inf 27.6%

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

   if -0.192 < m

   1. Initial program 87.5%

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

      1. sqr-neg 87.5%

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

      2. associate-+l+ 87.5%

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

      3. sqr-neg 87.5%

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

      4. distribute-rgt-out 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in m around 0 50.9%

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

      1. clear-num 50.6%

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

      2. +-commutative 50.6%

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

      3. +-commutative 50.6%

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

      4. fma-udef 50.6%

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

      5. inv-pow 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. unpow-1 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in k around 0 35.4%

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

   10. Taylor expanded in a around 0 35.4%

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

       1. *-commutative 35.4%

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

   12. Simplified 35.4%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.192:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot 10}{a}}\\ \end{array} \]

Alternative 6: 27.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 0.1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (* a (+ 1.0 (* k -10.0))) (/ (* a 0.1) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a * 0.1) / 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 <= 0.075d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (a * 0.1d0) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a * 0.1) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (a * 0.1) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a * 0.1) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (a * 0.1) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * 0.1), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   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. sqr-neg 98.7%

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

      2. associate-+l+ 98.7%

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

      3. sqr-neg 98.7%

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

      4. distribute-rgt-out 98.7%

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

   3. Simplified 98.7%

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

      1. expm1-log1p-u 72.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\right)\right)} \]

      2. expm1-udef 58.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\right)} - 1} \]

      3. associate-/l* 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{1 + k \cdot \left(10 + k\right)}{{k}^{m}}}}\right)} - 1 \]

      4. associate-/r/ 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)} \cdot {k}^{m}}\right)} - 1 \]

      5. +-commutative 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \cdot {k}^{m}\right)} - 1 \]

      6. fma-def 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot {k}^{m}\right)} - 1 \]

      7. +-commutative 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot {k}^{m}\right)} - 1 \]

   5. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot {k}^{m}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 72.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot {k}^{m}\right)\right)} \]

      2. expm1-log1p 98.0%

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

      3. associate-*l/ 98.7%

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

      4. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in m around 0 38.5%

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

   9. Taylor expanded in k around 0 36.1%

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

       1. *-commutative 36.1%

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

   11. Simplified 36.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. associate-+l+ 79.7%

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

      3. sqr-neg 79.7%

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

      4. distribute-rgt-out 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in m around 0 60.2%

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

      1. clear-num 59.9%

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

      2. +-commutative 59.9%

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

      3. +-commutative 59.9%

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

      4. fma-udef 59.9%

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

      5. inv-pow 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. unpow-1 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in k around 0 22.2%

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

   10. Taylor expanded in k around inf 21.4%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   11. Step-by-step derivation

       1. associate-*r/ 21.4%

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

   12. Simplified 21.4%

       \[\leadsto \color{blue}{\frac{0.1 \cdot a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 0.1}{k}\\ \end{array} \]

Alternative 7: 27.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (* a (+ 1.0 (* k -10.0))) (/ 1.0 (* 10.0 (/ k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 1.0 / (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 (k <= 0.075d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = 1.0d0 / (10.0d0 * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 1.0 / (10.0 * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = 1.0 / (10.0 * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(1.0 / Float64(10.0 * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = 1.0 / (10.0 * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(10.0 * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   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. sqr-neg 98.7%

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

      2. associate-+l+ 98.7%

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

      3. sqr-neg 98.7%

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

      4. distribute-rgt-out 98.7%

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

   3. Simplified 98.7%

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

      1. expm1-log1p-u 72.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\right)\right)} \]

      2. expm1-udef 58.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}\right)} - 1} \]

      3. associate-/l* 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{1 + k \cdot \left(10 + k\right)}{{k}^{m}}}}\right)} - 1 \]

      4. associate-/r/ 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)} \cdot {k}^{m}}\right)} - 1 \]

      5. +-commutative 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \cdot {k}^{m}\right)} - 1 \]

      6. fma-def 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot {k}^{m}\right)} - 1 \]

      7. +-commutative 58.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot {k}^{m}\right)} - 1 \]

   5. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot {k}^{m}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 72.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot {k}^{m}\right)\right)} \]

      2. expm1-log1p 98.0%

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

      3. associate-*l/ 98.7%

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

      4. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in m around 0 38.5%

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

   9. Taylor expanded in k around 0 36.1%

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

       1. *-commutative 36.1%

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

   11. Simplified 36.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. associate-+l+ 79.7%

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

      3. sqr-neg 79.7%

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

      4. distribute-rgt-out 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in m around 0 60.2%

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

      1. clear-num 59.9%

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

      2. +-commutative 59.9%

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

      3. +-commutative 59.9%

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

      4. fma-udef 59.9%

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

      5. inv-pow 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. unpow-1 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in k around 0 22.2%

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

   10. Taylor expanded in k around inf 22.2%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 8: 30.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.455:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.455) (/ 1.0 (* 10.0 (/ k a))) (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.455) {
		tmp = 1.0 / (10.0 * (k / a));
	} 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 <= (-0.455d0)) then
        tmp = 1.0d0 / (10.0d0 * (k / a))
    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 <= -0.455) {
		tmp = 1.0 / (10.0 * (k / a));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.455:
		tmp = 1.0 / (10.0 * (k / a))
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.455)
		tmp = Float64(1.0 / Float64(10.0 * Float64(k / a)));
	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 <= -0.455)
		tmp = 1.0 / (10.0 * (k / a));
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.455], N[(1.0 / N[(10.0 * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.455000000000000016

   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. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 38.2%

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

      1. clear-num 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-udef 38.3%

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

      5. inv-pow 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. unpow-1 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in k around 0 18.8%

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

   10. Taylor expanded in k around inf 27.6%

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

   if -0.455000000000000016 < m

   1. Initial program 87.5%

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

      1. sqr-neg 87.5%

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

      2. associate-+l+ 87.5%

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

      3. sqr-neg 87.5%

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

      4. distribute-rgt-out 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in m around 0 50.9%

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

   5. Taylor expanded in k around 0 35.2%

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

      1. *-commutative 35.2%

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

   7. Simplified 35.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.455:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 9: 44.5% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. sqr-neg 91.1%

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

   2. associate-+l+ 91.1%

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

   3. sqr-neg 91.1%

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

   4. distribute-rgt-out 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in m around 0 47.3%

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

5. Final simplification 47.3%

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

Alternative 10: 25.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 12500000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 12500000000.0) a (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 12500000000.0) {
		tmp = a;
	} else {
		tmp = 0.1 * (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 <= 12500000000.0d0) then
        tmp = a
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 12500000000.0) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 12500000000.0:
		tmp = a
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 12500000000.0)
		tmp = a;
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 12500000000.0)
		tmp = a;
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 12500000000.0], a, N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e10

   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. sqr-neg 98.7%

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

      2. associate-+l+ 98.7%

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

      3. sqr-neg 98.7%

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

      4. distribute-rgt-out 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in m around 0 38.3%

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

   5. Taylor expanded in k around 0 33.7%

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

   if 1.25e10 < k

   1. Initial program 79.5%

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

      1. sqr-neg 79.5%

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

      2. associate-+l+ 79.5%

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

      3. sqr-neg 79.5%

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

      4. distribute-rgt-out 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 60.8%

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

      1. clear-num 60.4%

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

      2. +-commutative 60.4%

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

      3. +-commutative 60.4%

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

      4. fma-udef 60.4%

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

      5. inv-pow 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. unpow-1 60.4%

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

   8. Simplified 60.4%

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

   9. Taylor expanded in k around 0 22.4%

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

   10. Taylor expanded in k around inf 21.6%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 12500000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 11: 25.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 12500000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 0.1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 12500000000.0) a (/ (* a 0.1) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 12500000000.0) {
		tmp = a;
	} else {
		tmp = (a * 0.1) / 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 <= 12500000000.0d0) then
        tmp = a
    else
        tmp = (a * 0.1d0) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 12500000000.0) {
		tmp = a;
	} else {
		tmp = (a * 0.1) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 12500000000.0:
		tmp = a
	else:
		tmp = (a * 0.1) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 12500000000.0)
		tmp = a;
	else
		tmp = Float64(Float64(a * 0.1) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 12500000000.0)
		tmp = a;
	else
		tmp = (a * 0.1) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 12500000000.0], a, N[(N[(a * 0.1), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e10

   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. sqr-neg 98.7%

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

      2. associate-+l+ 98.7%

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

      3. sqr-neg 98.7%

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

      4. distribute-rgt-out 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in m around 0 38.3%

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

   5. Taylor expanded in k around 0 33.7%

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

   if 1.25e10 < k

   1. Initial program 79.5%

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

      1. sqr-neg 79.5%

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

      2. associate-+l+ 79.5%

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

      3. sqr-neg 79.5%

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

      4. distribute-rgt-out 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in m around 0 60.8%

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

      1. clear-num 60.4%

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

      2. +-commutative 60.4%

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

      3. +-commutative 60.4%

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

      4. fma-udef 60.4%

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

      5. inv-pow 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. unpow-1 60.4%

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

   8. Simplified 60.4%

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

   9. Taylor expanded in k around 0 22.4%

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

   10. Taylor expanded in k around inf 21.6%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   11. Step-by-step derivation

       1. associate-*r/ 21.6%

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

   12. Simplified 21.6%

       \[\leadsto \color{blue}{\frac{0.1 \cdot a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 12500000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 0.1}{k}\\ \end{array} \]

Alternative 12: 19.7% 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 91.1%

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

   1. sqr-neg 91.1%

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

   2. associate-+l+ 91.1%

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

   3. sqr-neg 91.1%

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

   4. distribute-rgt-out 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in m around 0 47.3%

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

5. Taylor expanded in k around 0 21.9%

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

6. Final simplification 21.9%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023320 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))