Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 99.8%

Time: 4.3s

Alternatives: 13

Speedup: 1.1×

• 22.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\left(k - -10\right) - \frac{-1}{k}}}{k}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.5e-38)
   (/ (* a (pow k m)) 1.0)
   (/ (/ (* (pow k m) a) (- (- k -10.0) (/ -1.0 k))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.5e-38) {
		tmp = (a * pow(k, m)) / 1.0;
	} else {
		tmp = ((pow(k, m) * a) / ((k - -10.0) - (-1.0 / k))) / k;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (k <= 1.5d-38) then
        tmp = (a * (k ** m)) / 1.0d0
    else
        tmp = (((k ** m) * a) / ((k - (-10.0d0)) - ((-1.0d0) / k))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.5e-38) {
		tmp = (a * Math.pow(k, m)) / 1.0;
	} else {
		tmp = ((Math.pow(k, m) * a) / ((k - -10.0) - (-1.0 / k))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.5e-38:
		tmp = (a * math.pow(k, m)) / 1.0
	else:
		tmp = ((math.pow(k, m) * a) / ((k - -10.0) - (-1.0 / k))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.5e-38)
		tmp = Float64(Float64(a * (k ^ m)) / 1.0);
	else
		tmp = Float64(Float64(Float64((k ^ m) * a) / Float64(Float64(k - -10.0) - Float64(-1.0 / k))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.5e-38)
		tmp = (a * (k ^ m)) / 1.0;
	else
		tmp = (((k ^ m) * a) / ((k - -10.0) - (-1.0 / k))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.5e-38], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999994e-38

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

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

   if 1.49999999999999994e-38 < 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

Alternative 2: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+236}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* 10.0 k)) (* k k))) 5e+236)
     (* (pow k m) (/ a (fma (- k -10.0) k 1.0)))
     (/ t_0 1.0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (10.0 * k)) + (k * k))) <= 5e+236) {
		tmp = pow(k, m) * (a / fma((k - -10.0), k, 1.0));
	} else {
		tmp = t_0 / 1.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 5e+236)
		tmp = Float64((k ^ m) * Float64(a / fma(Float64(k - -10.0), k, 1.0)));
	else
		tmp = Float64(t_0 / 1.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+236], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 1.0), $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))) < 4.9999999999999997e236

   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 88.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. +-lft-identity N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. distribute-rgt-out N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 88.6%

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

   if 4.9999999999999997e236 < (/.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. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -1.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 62000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot \mathsf{fma}\left(\log k, m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -1.65)
     t_0
     (if (<= m 62000.0)
       (* (/ a (fma (- k -10.0) k 1.0)) (fma (log k) m 1.0))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -1.65) {
		tmp = t_0;
	} else if (m <= 62000.0) {
		tmp = (a / fma((k - -10.0), k, 1.0)) * fma(log(k), m, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -1.65)
		tmp = t_0;
	elseif (m <= 62000.0)
		tmp = Float64(Float64(a / fma(Float64(k - -10.0), k, 1.0)) * fma(log(k), m, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -1.65], t$95$0, If[LessEqual[m, 62000.0], N[(N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Log[k], $MachinePrecision] * m + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.6499999999999999 or 62000 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

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

   if -1.6499999999999999 < m < 62000

   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 88.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. +-lft-identity N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. distribute-rgt-out N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 40.9

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

   6. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-unsound-log.f64 N/A

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

      10. lower-unsound-*.f32 N/A

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

      11. lower-unsound-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-unsound-log.f64 40.9

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

   8. Applied rewrites 40.9%

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

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \mathbf{elif}\;m \leq 65000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.1e-9)
   (/ (pow k m) (/ 1.0 a))
   (if (<= m 65000.0) (/ a (fma (- k -10.0) k 1.0)) (/ (* a (pow k m)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-9) {
		tmp = pow(k, m) / (1.0 / a);
	} else if (m <= 65000.0) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = (a * pow(k, m)) / 1.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.1e-9)
		tmp = Float64((k ^ m) / Float64(1.0 / a));
	elseif (m <= 65000.0)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(a * (k ^ m)) / 1.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.1e-9], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 65000.0], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.10000000000000005e-9

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 82.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 82.2

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

   6. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. div-flip-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -3.10000000000000005e-9 < m < 65000

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-/.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + {k}^{0}} \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 44.5

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

   10. Applied rewrites 44.5%

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

   if 65000 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 65000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -3.1e-9)
     t_0
     (if (<= m 65000.0) (/ a (fma (- k -10.0) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -3.1e-9) {
		tmp = t_0;
	} else if (m <= 65000.0) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -3.1e-9)
		tmp = t_0;
	elseif (m <= 65000.0)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -3.1e-9], t$95$0, If[LessEqual[m, 65000.0], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -3.10000000000000005e-9 or 65000 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.2%

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

   if -3.10000000000000005e-9 < m < 65000

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-/.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + {k}^{0}} \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 44.5

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

   10. Applied rewrites 44.5%

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

Alternative 6: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left|a\right|}{k \cdot \left(\left(1 - \frac{\frac{-1}{k} - 10}{k}\right) \cdot k\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+285}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left|a\right|}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + -10 \cdot \left(\left|a\right| \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 0.0)
      (/ (fabs a) (* k (* (- 1.0 (/ (- (/ -1.0 k) 10.0) k)) k)))
      (if (<= t_0 1e+285)
        (/ (fabs a) (fma (- k -10.0) k 1.0))
        (if (<= t_0 INFINITY)
          (/ (fabs a) (pow k 2.0))
          (+ (fabs a) (* -10.0 (* (fabs a) k)))))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fabs(a) / (k * ((1.0 - (((-1.0 / k) - 10.0) / k)) * k));
	} else if (t_0 <= 1e+285) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs(a) / pow(k, 2.0);
	} else {
		tmp = fabs(a) + (-10.0 * (fabs(a) * k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(abs(a) / Float64(k * Float64(Float64(1.0 - Float64(Float64(Float64(-1.0 / k) - 10.0) / k)) * k)));
	elseif (t_0 <= 1e+285)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(abs(a) / (k ^ 2.0));
	else
		tmp = Float64(abs(a) + Float64(-10.0 * Float64(abs(a) * k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[(N[Abs[a], $MachinePrecision] / N[(k * N[(N[(1.0 - N[(N[(N[(-1.0 / k), $MachinePrecision] - 10.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+285], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Abs[a], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Abs[a], $MachinePrecision] + N[(-10.0 * N[(N[Abs[a], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 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))) < 0.0

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+r+ N/A

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

      4. add-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. associate--l- N/A

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

      13. +-commutative N/A

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

      14. sub-to-mult N/A

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

      15. lower-unsound-*.f64 N/A

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

      16. lower-unsound--.f64 N/A

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

      17. lower-unsound-/.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 45.2

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

   10. Applied rewrites 45.2%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999998e284

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-/.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + {k}^{0}} \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 44.5

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

   10. Applied rewrites 44.5%

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

   if 9.9999999999999998e284 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-pow.f64 36.7

          \[\leadsto \frac{a}{{k}^{2}} \]

   11. Applied rewrites 36.7%

       \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

   if +inf.0 < (/.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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. 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 20.1

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

   11. Applied rewrites 20.1%

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

Alternative 7: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(k - \frac{-1}{k}\right) - -10} \cdot \frac{1}{k}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.2e-9)
   (/ a (pow k 2.0))
   (* (/ a (- (- k (/ -1.0 k)) -10.0)) (/ 1.0 k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.2e-9) {
		tmp = a / pow(k, 2.0);
	} else {
		tmp = (a / ((k - (-1.0 / k)) - -10.0)) * (1.0 / k);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-3.2d-9)) then
        tmp = a / (k ** 2.0d0)
    else
        tmp = (a / ((k - ((-1.0d0) / k)) - (-10.0d0))) * (1.0d0 / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.2e-9) {
		tmp = a / Math.pow(k, 2.0);
	} else {
		tmp = (a / ((k - (-1.0 / k)) - -10.0)) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.2e-9:
		tmp = a / math.pow(k, 2.0)
	else:
		tmp = (a / ((k - (-1.0 / k)) - -10.0)) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.2e-9)
		tmp = Float64(a / (k ^ 2.0));
	else
		tmp = Float64(Float64(a / Float64(Float64(k - Float64(-1.0 / k)) - -10.0)) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.2e-9)
		tmp = a / (k ^ 2.0);
	else
		tmp = (a / ((k - (-1.0 / k)) - -10.0)) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.2e-9], N[(a / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(k - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.20000000000000012e-9

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-pow.f64 36.7

          \[\leadsto \frac{a}{{k}^{2}} \]

   11. Applied rewrites 36.7%

       \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

   if -3.20000000000000012e-9 < 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   10. Applied rewrites 44.5%

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

Alternative 8: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{\left|a\right|}{\left(k - \frac{-1}{k}\right) - -10}}{k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + -10 \cdot \left(\left|a\right| \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 0.0)
      (/ (/ (fabs a) (- (- k (/ -1.0 k)) -10.0)) k)
      (if (<= t_0 5e+236)
        (/ (fabs a) (fma (- k -10.0) k 1.0))
        (+ (fabs a) (* -10.0 (* (fabs a) k))))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fabs(a) / ((k - (-1.0 / k)) - -10.0)) / k;
	} else if (t_0 <= 5e+236) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fabs(a) + (-10.0 * (fabs(a) * k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(abs(a) / Float64(Float64(k - Float64(-1.0 / k)) - -10.0)) / k);
	elseif (t_0 <= 5e+236)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(abs(a) + Float64(-10.0 * Float64(abs(a) * k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[(N[(N[Abs[a], $MachinePrecision] / N[(N[(k - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$0, 5e+236], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[a], $MachinePrecision] + N[(-10.0 * N[(N[Abs[a], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $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))) < 0.0

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

   10. Applied rewrites 44.5%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999997e236

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-/.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + {k}^{0}} \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 44.5

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

   10. Applied rewrites 44.5%

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

   if 4.9999999999999997e236 < (/.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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. 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 20.1

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

   11. Applied rewrites 20.1%

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

Alternative 9: 48.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k - -10, k, 1\right) \cdot k}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\left(k - \frac{-1}{k}\right) - -10}}{k}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 6.2e-186)
   (/ a (/ (* (fma (- k -10.0) k 1.0) k) k))
   (/ (/ a (- (- k (/ -1.0 k)) -10.0)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.2e-186) {
		tmp = a / ((fma((k - -10.0), k, 1.0) * k) / k);
	} else {
		tmp = (a / ((k - (-1.0 / k)) - -10.0)) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6.2e-186)
		tmp = Float64(a / Float64(Float64(fma(Float64(k - -10.0), k, 1.0) * k) / k));
	else
		tmp = Float64(Float64(a / Float64(Float64(k - Float64(-1.0 / k)) - -10.0)) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6.2e-186], N[(a / N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(N[(k - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision] - -10.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.20000000000000018e-186

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-+r+ N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. sub-to-fraction N/A

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

      15. associate-*l/ N/A

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

      16. lower-/.f64 N/A

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

   10. Applied rewrites 44.7%

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

   if 6.20000000000000018e-186 < 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

   10. Applied rewrites 44.5%

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

Alternative 10: 46.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 65000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 65000.0) (/ a (fma (- k -10.0) k 1.0)) (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 65000.0) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 65000.0)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 65000.0], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 65000

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-/.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + {k}^{0}} \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 44.5

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

   10. Applied rewrites 44.5%

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

   if 65000 < 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. 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 20.1

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

   11. Applied rewrites 20.1%

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

Alternative 11: 29.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + 10 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 5e+15) (/ a (+ 1.0 (* 10.0 k))) (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5e+15) {
		tmp = a / (1.0 + (10.0 * k));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 5d+15) then
        tmp = a / (1.0d0 + (10.0d0 * k))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5e+15) {
		tmp = a / (1.0 + (10.0 * k));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5e+15:
		tmp = a / (1.0 + (10.0 * k))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5e+15)
		tmp = Float64(a / Float64(1.0 + Float64(10.0 * k)));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5e+15)
		tmp = a / (1.0 + (10.0 * k));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5e+15], N[(a / N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5e15

   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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.5

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 27.7

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

   11. Applied rewrites 27.7%

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

   if 5e15 < 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5

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

   3. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\left(k - -10\right) - \frac{-1}{k}}}{k}} \]

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \left(10 + \color{blue}{\left(k + \frac{1}{k}\right)}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \color{blue}{\frac{1}{k}}\right)\right)} \]

      5. lower-/.f64 44.5

         \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \frac{1}{\color{blue}{k}}\right)\right)} \]

   8. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   10. 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 20.1

          \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   11. Applied rewrites 20.1%

       \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 20.1% accurate, 3.6× speedup

Math

\[a + -10 \cdot \left(a \cdot k\right) \]

FPCore

(FPCore (a k m) :precision binary64 (+ a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	return a + (-10.0 * (a * 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 + ((-10.0d0) * (a * k))
end function

Java

public static double code(double a, double k, double m) {
	return a + (-10.0 * (a * k));
}

Python

def code(a, k, m):
	return a + (-10.0 * (a * k))

Julia

function code(a, k, m)
	return Float64(a + Float64(-10.0 * Float64(a * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = a + (-10.0 * (a * k));
end

Wolfram

code[a_, k_, m_] := N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $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

      \[\leadsto \frac{\color{blue}{a \cdot {k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. lower-*.f64 90.5

      \[\leadsto \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   5. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{k \cdot k + \color{blue}{\left(1 + 10 \cdot k\right)}} \]

   7. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{k \cdot k + \color{blue}{\left(10 \cdot k + 1\right)}} \]

   8. associate-+r+ N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(k \cdot k + 10 \cdot k\right) + 1}} \]

   9. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 \cdot k + k \cdot k\right)} + 1} \]

   10. metadata-eval N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(10 \cdot k + k \cdot k\right) + \color{blue}{\left(0 + 1\right)}} \]

   11. associate-+r+ N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(\left(10 \cdot k + k \cdot k\right) + 0\right) + 1}} \]

   12. +-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(0 + \left(10 \cdot k + k \cdot k\right)\right)} + 1} \]

   13. +-lft-identity N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 \cdot k + k \cdot k\right)} + 1} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(\color{blue}{10 \cdot k} + k \cdot k\right) + 1} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(10 \cdot k + \color{blue}{k \cdot k}\right) + 1} \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

   17. *-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 + k\right) \cdot k} + 1} \]

   18. lower-fma.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   19. +-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k + 10}, k, 1\right)} \]

   20. add-flip N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k - \left(\mathsf{neg}\left(10\right)\right)}, k, 1\right)} \]

   21. lower--.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k - \left(\mathsf{neg}\left(10\right)\right)}, k, 1\right)} \]

   22. metadata-eval 90.5

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k - \color{blue}{-10}, k, 1\right)} \]

3. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}} \]

5. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\left(k - -10\right) - \frac{-1}{k}}}{k}} \]

6. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{a}{k \cdot \left(10 + \color{blue}{\left(k + \frac{1}{k}\right)}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \color{blue}{\frac{1}{k}}\right)\right)} \]

   5. lower-/.f64 44.5

      \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \frac{1}{\color{blue}{k}}\right)\right)} \]

8. Applied rewrites 44.5%

   \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

9. Taylor expanded in k around 0

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
10. 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 20.1

       \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

11. Applied rewrites 20.1%

    \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
12. Add Preprocessing

Alternative 13: 19.1% accurate, 7.9× speedup

Math

\[\frac{a}{1} \]

FPCore

(FPCore (a k m) :precision binary64 (/ a 1.0))

C

double code(double a, double k, double m) {
	return a / 1.0;
}

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 / 1.0d0
end function

Java

public static double code(double a, double k, double m) {
	return a / 1.0;
}

Python

def code(a, k, m):
	return a / 1.0

Julia

function code(a, k, m)
	return Float64(a / 1.0)
end

MATLAB

function tmp = code(a, k, m)
	tmp = a / 1.0;
end

Wolfram

code[a_, k_, m_] := N[(a / 1.0), $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

      \[\leadsto \frac{\color{blue}{a \cdot {k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. lower-*.f64 90.5

      \[\leadsto \frac{\color{blue}{{k}^{m} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   5. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{k \cdot k + \color{blue}{\left(1 + 10 \cdot k\right)}} \]

   7. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{k \cdot k + \color{blue}{\left(10 \cdot k + 1\right)}} \]

   8. associate-+r+ N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(k \cdot k + 10 \cdot k\right) + 1}} \]

   9. +-commutative N/A

      \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 \cdot k + k \cdot k\right)} + 1} \]

   10. metadata-eval N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(10 \cdot k + k \cdot k\right) + \color{blue}{\left(0 + 1\right)}} \]

   11. associate-+r+ N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(\left(10 \cdot k + k \cdot k\right) + 0\right) + 1}} \]

   12. +-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(0 + \left(10 \cdot k + k \cdot k\right)\right)} + 1} \]

   13. +-lft-identity N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 \cdot k + k \cdot k\right)} + 1} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(\color{blue}{10 \cdot k} + k \cdot k\right) + 1} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\left(10 \cdot k + \color{blue}{k \cdot k}\right) + 1} \]

   16. distribute-rgt-out N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]

   17. *-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\left(10 + k\right) \cdot k} + 1} \]

   18. lower-fma.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\color{blue}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   19. +-commutative N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k + 10}, k, 1\right)} \]

   20. add-flip N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k - \left(\mathsf{neg}\left(10\right)\right)}, k, 1\right)} \]

   21. lower--.f64 N/A

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(\color{blue}{k - \left(\mathsf{neg}\left(10\right)\right)}, k, 1\right)} \]

   22. metadata-eval 90.5

       \[\leadsto \frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k - \color{blue}{-10}, k, 1\right)} \]

3. Applied rewrites 90.5%

   \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}} \]

5. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\left(k - -10\right) - \frac{-1}{k}}}{k}} \]

6. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{a}{k \cdot \left(10 + \color{blue}{\left(k + \frac{1}{k}\right)}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \color{blue}{\frac{1}{k}}\right)\right)} \]

   5. lower-/.f64 44.5

      \[\leadsto \frac{a}{k \cdot \left(10 + \left(k + \frac{1}{\color{blue}{k}}\right)\right)} \]

8. Applied rewrites 44.5%

   \[\leadsto \color{blue}{\frac{a}{k \cdot \left(10 + \left(k + \frac{1}{k}\right)\right)}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{a}{1} \]
10. Step-by-step derivation

11. Applied rewrites 19.1%

    \[\leadsto \frac{a}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))