Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 99.8%

Time: 6.3s

Alternatives: 13

Speedup: 1.3×

• 21.7% of points produce a very large (infinite) output. You may want to add a precondition.

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    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: 90.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-15}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\frac{1}{\left|k\right|}, \mathsf{fma}\left(10, k, 1\right), \left|k\right|\right)}}{\left|k\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 8e-15)
   (* a (pow k m))
   (/
    (/ (* (pow k m) a) (fma (/ 1.0 (fabs k)) (fma 10.0 k 1.0) (fabs k)))
    (fabs k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 8e-15) {
		tmp = a * pow(k, m);
	} else {
		tmp = ((pow(k, m) * a) / fma((1.0 / fabs(k)), fma(10.0, k, 1.0), fabs(k))) / fabs(k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 8e-15)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(Float64((k ^ m) * a) / fma(Float64(1.0 / abs(k)), fma(10.0, k, 1.0), abs(k))) / abs(k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 8e-15], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(1.0 / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(10.0 * k + 1.0), $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000006e-15

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 8.0000000000000006e-15 < k

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\frac{1}{\left|k\right|}, \mathsf{fma}\left(10, k, 1\right), \left|k\right|\right)}}{\left|k\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-30}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(\frac{1}{\left|k\right|}, \mathsf{fma}\left(10, k, 1\right), \left|k\right|\right)} \cdot \frac{a}{\left|k\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4e-30)
   (* a (pow k m))
   (*
    (/ (pow k m) (fma (/ 1.0 (fabs k)) (fma 10.0 k 1.0) (fabs k)))
    (/ a (fabs k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4e-30) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) / fma((1.0 / fabs(k)), fma(10.0, k, 1.0), fabs(k))) * (a / fabs(k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4e-30)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) / fma(Float64(1.0 / abs(k)), fma(10.0, k, 1.0), abs(k))) * Float64(a / abs(k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4e-30], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(10.0 * k + 1.0), $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4e-30

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 4e-30 < k

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\mathsf{fma}\left(\frac{1}{\left|k\right|}, \mathsf{fma}\left(10, k, 1\right), \left|k\right|\right)} \cdot \frac{a}{\left|k\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.175:\\ \;\;\;\;\frac{\frac{2}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot \left({k}^{m} \cdot a\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.175)
   (/ (* (/ 2.0 (fma (- k -10.0) k 1.0)) (* (pow k m) a)) 2.0)
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.175) {
		tmp = ((2.0 / fma((k - -10.0), k, 1.0)) * (pow(k, m) * a)) / 2.0;
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.175)
		tmp = Float64(Float64(Float64(2.0 / fma(Float64(k - -10.0), k, 1.0)) * Float64((k ^ m) * a)) / 2.0);
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.175], N[(N[(N[(2.0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.17499999999999999

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   if 0.17499999999999999 < m

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8

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

   12. Applied rewrites 82.8%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.175:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.175) (* (/ (pow k m) (fma (- k -10.0) k 1.0)) a) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.175) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.175)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.175], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.17499999999999999

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.5

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   if 0.17499999999999999 < m

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8

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

   12. Applied rewrites 82.8%

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

Alternative 5: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.82 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -1.6e-8)
     t_0
     (if (<= m 1.82e-6) (* (/ 1.0 (fma (- k -10.0) k 1.0)) a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -1.6e-8) {
		tmp = t_0;
	} else if (m <= 1.82e-6) {
		tmp = (1.0 / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -1.6e-8)
		tmp = t_0;
	elseif (m <= 1.82e-6)
		tmp = Float64(Float64(1.0 / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.6e-8], t$95$0, If[LessEqual[m, 1.82e-6], N[(N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.6000000000000001e-8 or 1.8199999999999999e-6 < m

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if -1.6000000000000001e-8 < m < 1.8199999999999999e-6

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 10^{-322}:\\ \;\;\;\;\frac{\frac{a}{\left|k\right| + \frac{1}{\left|k\right|}}}{\left|k\right|}\\ \mathbf{elif}\;t\_0 \leq 10^{+207}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 1e-322)
     (/ (/ a (+ (fabs k) (/ 1.0 (fabs k)))) (fabs k))
     (if (<= t_0 1e+207)
       (* (/ 1.0 (fma (- k -10.0) k 1.0)) a)
       (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a)))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 1e-322) {
		tmp = (a / (fabs(k) + (1.0 / fabs(k)))) / fabs(k);
	} else if (t_0 <= 1e+207) {
		tmp = (1.0 / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 1e-322)
		tmp = Float64(Float64(a / Float64(abs(k) + Float64(1.0 / abs(k)))) / abs(k));
	elseif (t_0 <= 1e+207)
		tmp = Float64(Float64(1.0 / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e-322], N[(N[(a / N[(N[Abs[k], $MachinePrecision] + N[(1.0 / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+207], N[(N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.88131e-323

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 44.6

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{a}{\left|k\right| + \color{blue}{\frac{1}{\left|k\right|}}}}{\left|k\right|} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\frac{a}{\left|k\right| + \frac{1}{\color{blue}{\left|k\right|}}}}{\left|k\right|} \]

       3. lower-fabs.f64 N/A

          \[\leadsto \frac{\frac{a}{\left|k\right| + \frac{1}{\left|\color{blue}{k}\right|}}}{\left|k\right|} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{a}{\left|k\right| + \frac{1}{\left|k\right|}}}{\left|k\right|} \]

       5. lower-fabs.f64 43.9

          \[\leadsto \frac{\frac{a}{\left|k\right| + \frac{1}{\left|k\right|}}}{\left|k\right|} \]

   11. Applied rewrites 43.9%

       \[\leadsto \frac{\frac{a}{\color{blue}{\left|k\right| + \frac{1}{\left|k\right|}}}}{\left|k\right|} \]

   if 9.88131e-323 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1e207

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if 1e207 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

Alternative 7: 53.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+207}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+207)
   (* (/ 1.0 (fma (- k -10.0) k 1.0)) a)
   (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+207) {
		tmp = (1.0 / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+207)
		tmp = Float64(Float64(1.0 / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 1e+207], N[(N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1e207

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if 1e207 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

       \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+287)
   (* (/ 1.0 (fma (- k -10.0) k 1.0)) a)
   (* k (fma -10.0 a (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+287) {
		tmp = (1.0 / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = k * fma(-10.0, a, (a / k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+287)
		tmp = Float64(Float64(1.0 / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = Float64(k * fma(-10.0, a, Float64(a / k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 2e+287], N[(N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(k * N[(-10.0 * a + N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.0000000000000002e287

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 44.8

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

   8. Applied rewrites 44.8%

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

   if 2.0000000000000002e287 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

       3. lower-/.f64 19.9

          \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   12. Applied rewrites 19.9%

       \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+287)
   (/ a (fma (- k -10.0) k 1.0))
   (* k (fma -10.0 a (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+287) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * fma(-10.0, a, (a / k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+287)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * fma(-10.0, a, Float64(a / k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 2e+287], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * a + N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.0000000000000002e287

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. sub-flip N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 44.9

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

   8. Applied rewrites 44.9%

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

   if 2.0000000000000002e287 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

   10. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

       3. lower-/.f64 19.9

          \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   12. Applied rewrites 19.9%

       \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 44.9% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (a k m) :precision binary64 (/ a (fma (- k -10.0) k 1.0)))

C

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

Julia

function code(a, k, m)
	return Float64(a / fma(Float64(k - -10.0), k, 1.0))
end

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 90.4%

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

4. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 44.9

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

6. Applied rewrites 44.9%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-flip N/A

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

   6. lift--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f64 44.9

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

8. Applied rewrites 44.9%

   \[\leadsto \frac{a}{\mathsf{fma}\left(k - -10, \color{blue}{k}, 1\right)} \]
9. Add Preprocessing

Alternative 11: 29.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.8e+33) (/ a (+ 1.0 (* k 10.0))) (* (fma -10.0 k 1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.8e+33) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = fma(-10.0, k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.8e+33)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(fma(-10.0, k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.8e+33], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.8000000000000001e33

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 28.1%

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

   if 2.8000000000000001e33 < m

   1. Initial program 90.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 44.9

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

   6. Applied rewrites 44.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.7

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   9. Applied rewrites 20.7%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

          \[\leadsto a + \left(a \cdot k\right) \cdot -10 \]

       4. fp-cancel-sign-sub-inv N/A

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

       5. *-commutative N/A

          \[\leadsto a - -10 \cdot \left(\mathsf{neg}\left(a \cdot k\right)\right) \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto a - \left(\mathsf{neg}\left(-10 \cdot \left(a \cdot k\right)\right)\right) \]

       7. distribute-lft-neg-out N/A

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

       8. metadata-eval N/A

          \[\leadsto a - 10 \cdot \left(a \cdot k\right) \]

       9. lift-*.f64 N/A

          \[\leadsto a - 10 \cdot \left(a \cdot k\right) \]

       10. *-commutative N/A

           \[\leadsto a - 10 \cdot \left(k \cdot a\right) \]

       11. associate-*r* N/A

           \[\leadsto a - \left(10 \cdot k\right) \cdot a \]

       12. fp-cancel-sub-sign-inv N/A

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

       13. distribute-rgt1-in N/A

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

       14. lower-*.f64 N/A

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

       15. distribute-lft-neg-out N/A

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

       16. metadata-eval N/A

           \[\leadsto \left(-10 \cdot k + 1\right) \cdot a \]

       17. lower-fma.f64 20.7

           \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]

   11. Applied rewrites 20.7%

       \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 20.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-10 \cdot a, k, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma (* -10.0 a) k a))

C

double code(double a, double k, double m) {
	return fma((-10.0 * a), k, a);
}

Julia

function code(a, k, m)
	return fma(Float64(-10.0 * a), k, a)
end

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 90.4%

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

4. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 44.9

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

6. Applied rewrites 44.9%

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

7. Taylor expanded in k around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 20.7

      \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

9. Applied rewrites 20.7%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

       \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

    3. add-flip N/A

       \[\leadsto -10 \cdot \left(a \cdot k\right) - \left(\mathsf{neg}\left(a\right)\right) \]

    4. sub-flip N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. remove-double-neg N/A

       \[\leadsto \left(-10 \cdot a\right) \cdot k + a \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-10 \cdot a, k, a\right) \]

    10. lower-*.f64 20.7

        \[\leadsto \mathsf{fma}\left(-10 \cdot a, k, a\right) \]

11. Applied rewrites 20.7%

    \[\leadsto \mathsf{fma}\left(-10 \cdot a, k, a\right) \]
12. Add Preprocessing

Alternative 13: 20.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 90.4%

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

4. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 44.9

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

6. Applied rewrites 44.9%

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

7. Taylor expanded in k around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 20.7

      \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

9. Applied rewrites 20.7%

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

    1. lift-+.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

       \[\leadsto a + \left(a \cdot k\right) \cdot -10 \]

    4. fp-cancel-sign-sub-inv N/A

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

    5. *-commutative N/A

       \[\leadsto a - -10 \cdot \left(\mathsf{neg}\left(a \cdot k\right)\right) \]

    6. distribute-rgt-neg-in N/A

       \[\leadsto a - \left(\mathsf{neg}\left(-10 \cdot \left(a \cdot k\right)\right)\right) \]

    7. distribute-lft-neg-out N/A

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

    8. metadata-eval N/A

       \[\leadsto a - 10 \cdot \left(a \cdot k\right) \]

    9. lift-*.f64 N/A

       \[\leadsto a - 10 \cdot \left(a \cdot k\right) \]

    10. *-commutative N/A

        \[\leadsto a - 10 \cdot \left(k \cdot a\right) \]

    11. associate-*r* N/A

        \[\leadsto a - \left(10 \cdot k\right) \cdot a \]

    12. fp-cancel-sub-sign-inv N/A

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

    13. distribute-rgt1-in N/A

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

    14. lower-*.f64 N/A

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

    15. distribute-lft-neg-out N/A

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

    16. metadata-eval N/A

        \[\leadsto \left(-10 \cdot k + 1\right) \cdot a \]

    17. lower-fma.f64 20.7

        \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]

11. Applied rewrites 20.7%

    \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))