Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.7%

Time: 5.5s

Alternatives: 14

Speedup: 1.1×

• 78.3% 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: 90.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := {k}^{\left(-m\right)}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{a} + \left(k - -10\right) \cdot \frac{k}{a \cdot {k}^{m}}}\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (let* ((t_0 (pow k (- m))))
  (if (<= k 5e-95)
    (/ a (* 1.0 t_0))
    (/ 1.0 (+ (/ t_0 a) (* (- k -10.0) (/ k (* a (pow k m)))))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, -m);
	double tmp;
	if (k <= 5e-95) {
		tmp = a / (1.0 * t_0);
	} else {
		tmp = 1.0 / ((t_0 / a) + ((k - -10.0) * (k / (a * pow(k, m)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = k ** -m
    if (k <= 5d-95) then
        tmp = a / (1.0d0 * t_0)
    else
        tmp = 1.0d0 / ((t_0 / a) + ((k - (-10.0d0)) * (k / (a * (k ** m)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, -m);
	double tmp;
	if (k <= 5e-95) {
		tmp = a / (1.0 * t_0);
	} else {
		tmp = 1.0 / ((t_0 / a) + ((k - -10.0) * (k / (a * Math.pow(k, m)))));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, -m)
	tmp = 0
	if k <= 5e-95:
		tmp = a / (1.0 * t_0)
	else:
		tmp = 1.0 / ((t_0 / a) + ((k - -10.0) * (k / (a * math.pow(k, m)))))
	return tmp

Julia

function code(a, k, m)
	t_0 = k ^ Float64(-m)
	tmp = 0.0
	if (k <= 5e-95)
		tmp = Float64(a / Float64(1.0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 / a) + Float64(Float64(k - -10.0) * Float64(k / Float64(a * (k ^ m))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k ^ -m;
	tmp = 0.0;
	if (k <= 5e-95)
		tmp = a / (1.0 * t_0);
	else
		tmp = 1.0 / ((t_0 / a) + ((k - -10.0) * (k / (a * (k ^ m)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999998e-95

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. mult-flip N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 82.8%

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

   if 4.9999999999999998e-95 < k

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 88.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;\frac{a}{\left(\left(k - -10\right) \cdot k - -1\right) \cdot {k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (if (<= m -1.35e-25)
  (/ a (* (- (* (- k -10.0) k) -1.0) (pow k (- m))))
  (if (<= m 1.85e-7)
    (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
    (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.35e-25) {
		tmp = a / ((((k - -10.0) * k) - -1.0) * pow(k, -m));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a * pow(k, m);
	}
	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 <= (-1.35d-25)) then
        tmp = a / ((((k - (-10.0d0)) * k) - (-1.0d0)) * (k ** -m))
    else if (m <= 1.85d-7) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.35e-25) {
		tmp = a / ((((k - -10.0) * k) - -1.0) * Math.pow(k, -m));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.35e-25:
		tmp = a / ((((k - -10.0) * k) - -1.0) * math.pow(k, -m))
	elif m <= 1.85e-7:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.35e-25)
		tmp = Float64(a / Float64(Float64(Float64(Float64(k - -10.0) * k) - -1.0) * (k ^ Float64(-m))));
	elseif (m <= 1.85e-7)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.35e-25)
		tmp = a / ((((k - -10.0) * k) - -1.0) * (k ^ -m));
	elseif (m <= 1.85e-7)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.35e-25], N[(a / N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k), $MachinePrecision] - -1.0), $MachinePrecision] * N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85e-7], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3500000000000001e-25

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. mult-flip N/A

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

   5. Applied rewrites 90.5%

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

   if -1.3500000000000001e-25 < m < 1.85e-7

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

   if 1.85e-7 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;\frac{t\_0}{1 + k \cdot k}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (let* ((t_0 (* a (pow k m))))
  (if (<= m -1.35e-25)
    (/ t_0 (+ 1.0 (* k k)))
    (if (<= m 1.85e-7)
      (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
      t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -1.35e-25) {
		tmp = t_0 / (1.0 + (k * k));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-1.35d-25)) then
        tmp = t_0 / (1.0d0 + (k * k))
    else if (m <= 1.85d-7) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -1.35e-25) {
		tmp = t_0 / (1.0 + (k * k));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -1.35e-25:
		tmp = t_0 / (1.0 + (k * k))
	elif m <= 1.85e-7:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -1.35e-25)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * k)));
	elseif (m <= 1.85e-7)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -1.35e-25)
		tmp = t_0 / (1.0 + (k * k));
	elseif (m <= 1.85e-7)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.35e-25], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85e-7], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3500000000000001e-25

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 89.6%

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

   if -1.3500000000000001e-25 < m < 1.85e-7

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

   if 1.85e-7 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 4: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 \cdot {k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (if (<= m -2.15e-13)
  (/ a (* 1.0 (pow k (- m))))
  (if (<= m 1.85e-7)
    (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
    (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.15e-13) {
		tmp = a / (1.0 * pow(k, -m));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a * pow(k, m);
	}
	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 <= (-2.15d-13)) then
        tmp = a / (1.0d0 * (k ** -m))
    else if (m <= 1.85d-7) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.15e-13) {
		tmp = a / (1.0 * Math.pow(k, -m));
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.15e-13:
		tmp = a / (1.0 * math.pow(k, -m))
	elif m <= 1.85e-7:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.15e-13)
		tmp = Float64(a / Float64(1.0 * (k ^ Float64(-m))));
	elseif (m <= 1.85e-7)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.15e-13)
		tmp = a / (1.0 * (k ^ -m));
	elseif (m <= 1.85e-7)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.15e-13], N[(a / N[(1.0 * N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85e-7], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.1499999999999999e-13

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. mult-flip N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 82.8%

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

   if -2.1499999999999999e-13 < m < 1.85e-7

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

   if 1.85e-7 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8%

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

   12. Applied rewrites 82.8%

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

Alternative 5: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.15 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (let* ((t_0 (* a (pow k m))))
  (if (<= m -2.15e-13)
    t_0
    (if (<= m 1.85e-7)
      (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
      t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -2.15e-13) {
		tmp = t_0;
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= (-2.15d-13)) then
        tmp = t_0
    else if (m <= 1.85d-7) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= -2.15e-13) {
		tmp = t_0;
	} else if (m <= 1.85e-7) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= -2.15e-13:
		tmp = t_0
	elif m <= 1.85e-7:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -2.15e-13)
		tmp = t_0;
	elseif (m <= 1.85e-7)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= -2.15e-13)
		tmp = t_0;
	elseif (m <= 1.85e-7)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.15e-13], t$95$0, If[LessEqual[m, 1.85e-7], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.1499999999999999e-13 or 1.85e-7 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 82.8%

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

   12. Applied rewrites 82.8%

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

   if -2.1499999999999999e-13 < m < 1.85e-7

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

Alternative 6: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ t_1 := \frac{1}{\left|a\right|}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{k \cdot \left(k \cdot \left(t\_1 + 10 \cdot \frac{1}{\left|a\right| \cdot k}\right)\right) + t\_1}\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;\frac{1}{1 + k \cdot \left(10 + k\right)} \cdot \left|a\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\left(\left(k - -10\right) \cdot k\right) \cdot \left|a\right| - \left|a\right| \cdot -1}{\left|a\right| \cdot \left|a\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + k \cdot \left(-1 \cdot \left(k \cdot \left(\left|a\right| + -100 \cdot \left|a\right|\right)\right) - 10 \cdot \left|a\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
       (t_1 (/ 1.0 (fabs a))))
  (*
   (copysign 1.0 a)
   (if (<= t_0 0.0)
     (/
      1.0
      (+ (* k (* k (+ t_1 (* 10.0 (/ 1.0 (* (fabs a) k)))))) t_1))
     (if (<= t_0 1e+298)
       (* (/ 1.0 (+ 1.0 (* k (+ 10.0 k)))) (fabs a))
       (if (<= t_0 INFINITY)
         (/
          1.0
          (/
           (- (* (* (- k -10.0) k) (fabs a)) (* (fabs a) -1.0))
           (* (fabs a) (fabs a))))
         (+
          (fabs a)
          (*
           k
           (-
            (* -1.0 (* k (+ (fabs a) (* -100.0 (fabs a)))))
            (* 10.0 (fabs a)))))))))))

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 t_1 = 1.0 / fabs(a);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / ((k * (k * (t_1 + (10.0 * (1.0 / (fabs(a) * k)))))) + t_1);
	} else if (t_0 <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * fabs(a);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / (((((k - -10.0) * k) * fabs(a)) - (fabs(a) * -1.0)) / (fabs(a) * fabs(a)));
	} else {
		tmp = fabs(a) + (k * ((-1.0 * (k * (fabs(a) + (-100.0 * fabs(a))))) - (10.0 * fabs(a))));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = (Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double t_1 = 1.0 / Math.abs(a);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 / ((k * (k * (t_1 + (10.0 * (1.0 / (Math.abs(a) * k)))))) + t_1);
	} else if (t_0 <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * Math.abs(a);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (((((k - -10.0) * k) * Math.abs(a)) - (Math.abs(a) * -1.0)) / (Math.abs(a) * Math.abs(a)));
	} else {
		tmp = Math.abs(a) + (k * ((-1.0 * (k * (Math.abs(a) + (-100.0 * Math.abs(a))))) - (10.0 * Math.abs(a))));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = (math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	t_1 = 1.0 / math.fabs(a)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 / ((k * (k * (t_1 + (10.0 * (1.0 / (math.fabs(a) * k)))))) + t_1)
	elif t_0 <= 1e+298:
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * math.fabs(a)
	elif t_0 <= math.inf:
		tmp = 1.0 / (((((k - -10.0) * k) * math.fabs(a)) - (math.fabs(a) * -1.0)) / (math.fabs(a) * math.fabs(a)))
	else:
		tmp = math.fabs(a) + (k * ((-1.0 * (k * (math.fabs(a) + (-100.0 * math.fabs(a))))) - (10.0 * math.fabs(a))))
	return math.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)))
	t_1 = Float64(1.0 / abs(a))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(k * Float64(k * Float64(t_1 + Float64(10.0 * Float64(1.0 / Float64(abs(a) * k)))))) + t_1));
	elseif (t_0 <= 1e+298)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(k * Float64(10.0 + k)))) * abs(a));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(k - -10.0) * k) * abs(a)) - Float64(abs(a) * -1.0)) / Float64(abs(a) * abs(a))));
	else
		tmp = Float64(abs(a) + Float64(k * Float64(Float64(-1.0 * Float64(k * Float64(abs(a) + Float64(-100.0 * abs(a))))) - Float64(10.0 * abs(a)))));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	t_1 = 1.0 / abs(a);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 / ((k * (k * (t_1 + (10.0 * (1.0 / (abs(a) * k)))))) + t_1);
	elseif (t_0 <= 1e+298)
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * abs(a);
	elseif (t_0 <= Inf)
		tmp = 1.0 / (((((k - -10.0) * k) * abs(a)) - (abs(a) * -1.0)) / (abs(a) * abs(a)));
	else
		tmp = abs(a) + (k * ((-1.0 * (k * (abs(a) + (-100.0 * abs(a))))) - (10.0 * abs(a))));
	end
	tmp_2 = (sign(a) * abs(1.0)) * 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]}, Block[{t$95$1 = N[(1.0 / N[Abs[a], $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[(1.0 / N[(N[(k * N[(k * N[(t$95$1 + N[(10.0 * N[(1.0 / N[(N[Abs[a], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(N[(1.0 / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] - N[(N[Abs[a], $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[a], $MachinePrecision] + N[(k * N[(N[(-1.0 * N[(k * N[(N[Abs[a], $MachinePrecision] + N[(-100.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

   10. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 46.0%

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

   12. Applied rewrites 46.0%

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

   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))) < 9.9999999999999996e297

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.1%

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

   6. Applied rewrites 46.1%

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

   if 9.9999999999999996e297 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. sub-div N/A

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

      13. frac-sub N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 40.3%

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

   8. Applied rewrites 40.3%

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

   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 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - \color{blue}{10 \cdot a}\right) \]

       3. lower--.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot \color{blue}{a}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       6. lower-+.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       8. lower-*.f64 27.8%

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   12. Applied rewrites 27.8%

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(k - -10\right) \cdot k\right) \cdot a - a \cdot -1}{a \cdot a}}\\ \mathbf{elif}\;m \leq 6.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (if (<= m -2.5e+17)
  (/ 1.0 (/ (- (* (* (- k -10.0) k) a) (* a -1.0)) (* a a)))
  (if (<= m 6.4e-5)
    (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
    (+ a (* k (- (* -1.0 (* k (+ a (* -100.0 a)))) (* 10.0 a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.5e+17) {
		tmp = 1.0 / (((((k - -10.0) * k) * a) - (a * -1.0)) / (a * a));
	} else if (m <= 6.4e-5) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	}
	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 <= (-2.5d+17)) then
        tmp = 1.0d0 / (((((k - (-10.0d0)) * k) * a) - (a * (-1.0d0))) / (a * a))
    else if (m <= 6.4d-5) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = a + (k * (((-1.0d0) * (k * (a + ((-100.0d0) * a)))) - (10.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.5e+17) {
		tmp = 1.0 / (((((k - -10.0) * k) * a) - (a * -1.0)) / (a * a));
	} else if (m <= 6.4e-5) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.5e+17:
		tmp = 1.0 / (((((k - -10.0) * k) * a) - (a * -1.0)) / (a * a))
	elif m <= 6.4e-5:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.5e+17)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(k - -10.0) * k) * a) - Float64(a * -1.0)) / Float64(a * a)));
	elseif (m <= 6.4e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = Float64(a + Float64(k * Float64(Float64(-1.0 * Float64(k * Float64(a + Float64(-100.0 * a)))) - Float64(10.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.5e+17)
		tmp = 1.0 / (((((k - -10.0) * k) * a) - (a * -1.0)) / (a * a));
	elseif (m <= 6.4e-5)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.5e+17], N[(1.0 / N[(N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k), $MachinePrecision] * a), $MachinePrecision] - N[(a * -1.0), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.4e-5], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(N[(-1.0 * N[(k * N[(a + N[(-100.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(10.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.5e17

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. sub-flip N/A

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

      12. sub-div N/A

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

      13. frac-sub N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 40.3%

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

   8. Applied rewrites 40.3%

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

   if -2.5e17 < m < 6.3999999999999997e-5

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

   if 6.3999999999999997e-5 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - \color{blue}{10 \cdot a}\right) \]

       3. lower--.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot \color{blue}{a}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       6. lower-+.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       8. lower-*.f64 27.8%

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   12. Applied rewrites 27.8%

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -9 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{\left(10 \cdot k\right) \cdot a + a \cdot 1}{a \cdot a}}\\ \mathbf{elif}\;m \leq 6.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (if (<= m -9e+23)
  (/ 1.0 (/ (+ (* (* 10.0 k) a) (* a 1.0)) (* a a)))
  (if (<= m 6.4e-5)
    (/ 1.0 (- (* (/ (- k -10.0) a) k) (/ -1.0 a)))
    (+ a (* k (- (* -1.0 (* k (+ a (* -100.0 a)))) (* 10.0 a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+23) {
		tmp = 1.0 / ((((10.0 * k) * a) + (a * 1.0)) / (a * a));
	} else if (m <= 6.4e-5) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	}
	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 <= (-9d+23)) then
        tmp = 1.0d0 / ((((10.0d0 * k) * a) + (a * 1.0d0)) / (a * a))
    else if (m <= 6.4d-5) then
        tmp = 1.0d0 / ((((k - (-10.0d0)) / a) * k) - ((-1.0d0) / a))
    else
        tmp = a + (k * (((-1.0d0) * (k * (a + ((-100.0d0) * a)))) - (10.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -9e+23) {
		tmp = 1.0 / ((((10.0 * k) * a) + (a * 1.0)) / (a * a));
	} else if (m <= 6.4e-5) {
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	} else {
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -9e+23:
		tmp = 1.0 / ((((10.0 * k) * a) + (a * 1.0)) / (a * a))
	elif m <= 6.4e-5:
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a))
	else:
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9e+23)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(10.0 * k) * a) + Float64(a * 1.0)) / Float64(a * a)));
	elseif (m <= 6.4e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / a) * k) - Float64(-1.0 / a)));
	else
		tmp = Float64(a + Float64(k * Float64(Float64(-1.0 * Float64(k * Float64(a + Float64(-100.0 * a)))) - Float64(10.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -9e+23)
		tmp = 1.0 / ((((10.0 * k) * a) + (a * 1.0)) / (a * a));
	elseif (m <= 6.4e-5)
		tmp = 1.0 / ((((k - -10.0) / a) * k) - (-1.0 / a));
	else
		tmp = a + (k * ((-1.0 * (k * (a + (-100.0 * a)))) - (10.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9e+23], N[(1.0 / N[(N[(N[(N[(10.0 * k), $MachinePrecision] * a), $MachinePrecision] + N[(a * 1.0), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.4e-5], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(N[(-1.0 * N[(k * N[(a + N[(-100.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(10.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.9999999999999996e23

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 29.4%

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

   9. Applied rewrites 29.4%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. div-add N/A

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

       5. frac-add N/A

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

       6. lower-/.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 31.9%

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

   11. Applied rewrites 31.9%

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

   if -8.9999999999999996e23 < m < 6.3999999999999997e-5

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 90.3%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 90.3%

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.0%

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

   9. Applied rewrites 46.0%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. mult-flip-rev N/A

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

       11. lift-/.f64 N/A

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

       12. div-add-rev N/A

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

       13. +-commutative N/A

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

       14. metadata-eval N/A

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

       15. sub-flip N/A

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

       16. lift--.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

       21. lower-/.f64 46.0%

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

   11. Applied rewrites 46.0%

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

   if 6.3999999999999997e-5 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

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

   7. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around 0

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - \color{blue}{10 \cdot a}\right) \]

       3. lower--.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot \color{blue}{a}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       6. lower-+.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       8. lower-*.f64 27.8%

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   12. Applied rewrites 27.8%

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;\frac{1}{\frac{k - -10}{\left|a\right|} \cdot k - \frac{-1}{\left|a\right|}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right| + \frac{\left|a\right|}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + k \cdot \left(-1 \cdot \left(k \cdot \left(\left|a\right| + -100 \cdot \left|a\right|\right)\right) - 10 \cdot \left|a\right|\right)\\ \end{array} \end{array} \]

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 1e+298)
     (/ 1.0 (- (* (/ (- k -10.0) (fabs a)) k) (/ -1.0 (fabs a))))
     (if (<= t_0 INFINITY)
       (* k (+ (* -10.0 (fabs a)) (/ (fabs a) k)))
       (+
        (fabs a)
        (*
         k
         (-
          (* -1.0 (* k (+ (fabs a) (* -100.0 (fabs a)))))
          (* 10.0 (fabs a))))))))))

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 <= 1e+298) {
		tmp = 1.0 / ((((k - -10.0) / fabs(a)) * k) - (-1.0 / fabs(a)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = k * ((-10.0 * fabs(a)) + (fabs(a) / k));
	} else {
		tmp = fabs(a) + (k * ((-1.0 * (k * (fabs(a) + (-100.0 * fabs(a))))) - (10.0 * fabs(a))));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = (Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 1e+298) {
		tmp = 1.0 / ((((k - -10.0) / Math.abs(a)) * k) - (-1.0 / Math.abs(a)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = k * ((-10.0 * Math.abs(a)) + (Math.abs(a) / k));
	} else {
		tmp = Math.abs(a) + (k * ((-1.0 * (k * (Math.abs(a) + (-100.0 * Math.abs(a))))) - (10.0 * Math.abs(a))));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = (math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_0 <= 1e+298:
		tmp = 1.0 / ((((k - -10.0) / math.fabs(a)) * k) - (-1.0 / math.fabs(a)))
	elif t_0 <= math.inf:
		tmp = k * ((-10.0 * math.fabs(a)) + (math.fabs(a) / k))
	else:
		tmp = math.fabs(a) + (k * ((-1.0 * (k * (math.fabs(a) + (-100.0 * math.fabs(a))))) - (10.0 * math.fabs(a))))
	return math.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 <= 1e+298)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k - -10.0) / abs(a)) * k) - Float64(-1.0 / abs(a))));
	elseif (t_0 <= Inf)
		tmp = Float64(k * Float64(Float64(-10.0 * abs(a)) + Float64(abs(a) / k)));
	else
		tmp = Float64(abs(a) + Float64(k * Float64(Float64(-1.0 * Float64(k * Float64(abs(a) + Float64(-100.0 * abs(a))))) - Float64(10.0 * abs(a)))));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_0 <= 1e+298)
		tmp = 1.0 / ((((k - -10.0) / abs(a)) * k) - (-1.0 / abs(a)));
	elseif (t_0 <= Inf)
		tmp = k * ((-10.0 * abs(a)) + (abs(a) / k));
	else
		tmp = abs(a) + (k * ((-1.0 * (k * (abs(a) + (-100.0 * abs(a))))) - (10.0 * abs(a))));
	end
	tmp_2 = (sign(a) * abs(1.0)) * 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, 1e+298], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] - N[(-1.0 / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(k * N[(N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[a], $MachinePrecision] + N[(k * N[(N[(-1.0 * N[(k * N[(N[Abs[a], $MachinePrecision] + N[(-100.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999996e297

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 90.3%

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot \left(10 + k\right)} - -1}{a \cdot {k}^{m}}} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(10 + k\right) \cdot k} - -1}{a \cdot {k}^{m}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(10 + k\right) \cdot k} - -1}{a \cdot {k}^{m}}} \]

      17. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(k + 10\right)} \cdot k - -1}{a \cdot {k}^{m}}} \]

      18. add-flip N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(k - \left(\mathsf{neg}\left(10\right)\right)\right)} \cdot k - -1}{a \cdot {k}^{m}}} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(k - \left(\mathsf{neg}\left(10\right)\right)\right)} \cdot k - -1}{a \cdot {k}^{m}}} \]

      20. metadata-eval 90.3%

          \[\leadsto \frac{1}{\frac{\left(k - \color{blue}{-10}\right) \cdot k - -1}{a \cdot {k}^{m}}} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k - -10\right) \cdot k - -1}{\color{blue}{a \cdot {k}^{m}}}} \]

      22. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k - -10\right) \cdot k - -1}{\color{blue}{{k}^{m} \cdot a}}} \]

      23. lower-*.f64 90.3%

          \[\leadsto \frac{1}{\frac{\left(k - -10\right) \cdot k - -1}{\color{blue}{{k}^{m} \cdot a}}} \]

   3. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k - -10\right) \cdot k - -1}{{k}^{m} \cdot a}}} \]

   4. Taylor expanded in m around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + k \cdot \left(10 + k\right)}{a}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1 + k \cdot \left(10 + k\right)}{\color{blue}{a}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1 + k \cdot \left(10 + k\right)}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 + k \cdot \left(10 + k\right)}{a}} \]

      4. lower-+.f64 46.0%

         \[\leadsto \frac{1}{\frac{1 + k \cdot \left(10 + k\right)}{a}} \]

   6. Applied rewrites 46.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + k \cdot \left(10 + k\right)}{a}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \color{blue}{\frac{1}{a}}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{\color{blue}{a}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

      7. lower-/.f64 46.0%

         \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{a}} \]

   9. Applied rewrites 46.0%

      \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \color{blue}{\frac{1}{a}}} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) + \frac{1}{\color{blue}{a}}} \]

       2. add-flip N/A

          \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{k \cdot \left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{\left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{1}{\left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       7. lift-+.f64 N/A

          \[\leadsto \frac{1}{\left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(10 \cdot \frac{1}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       10. mult-flip-rev N/A

           \[\leadsto \frac{1}{\left(\frac{10}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{1}{\left(\frac{10}{a} + \frac{k}{a}\right) \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       12. div-add-rev N/A

           \[\leadsto \frac{1}{\frac{10 + k}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       13. +-commutative N/A

           \[\leadsto \frac{1}{\frac{k + 10}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       14. metadata-eval N/A

           \[\leadsto \frac{1}{\frac{k + \left(\mathsf{neg}\left(-10\right)\right)}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       15. sub-flip N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       16. lift--.f64 N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       17. lower-/.f64 N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       18. lift-/.f64 N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

       19. distribute-neg-frac N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \frac{\mathsf{neg}\left(1\right)}{a}} \]

       20. metadata-eval N/A

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}} \]

       21. lower-/.f64 46.0%

           \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{a}} \]

   11. Applied rewrites 46.0%

       \[\leadsto \frac{1}{\frac{k - -10}{a} \cdot k - \frac{-1}{\color{blue}{a}}} \]

   if 9.9999999999999996e297 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      4. lower-/.f64 20.2%

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

   10. Applied rewrites 20.2%

       \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]

   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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

      3. associate-*r* N/A

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

      4. lower-*.f64 N/A

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

      5. lower-*.f64 21.5%

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

   9. Applied rewrites 21.5%

      \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

   10. Taylor expanded in k around 0

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - \color{blue}{10 \cdot a}\right) \]

       3. lower--.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot \color{blue}{a}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       6. lower-+.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       8. lower-*.f64 27.8%

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   12. Applied rewrites 27.8%

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;\frac{1}{1 + k \cdot \left(10 + k\right)} \cdot \left|a\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right| + \frac{\left|a\right|}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + k \cdot \left(-1 \cdot \left(k \cdot \left(\left|a\right| + -100 \cdot \left|a\right|\right)\right) - 10 \cdot \left|a\right|\right)\\ \end{array} \end{array} \]

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 1e+298)
     (* (/ 1.0 (+ 1.0 (* k (+ 10.0 k)))) (fabs a))
     (if (<= t_0 INFINITY)
       (* k (+ (* -10.0 (fabs a)) (/ (fabs a) k)))
       (+
        (fabs a)
        (*
         k
         (-
          (* -1.0 (* k (+ (fabs a) (* -100.0 (fabs a)))))
          (* 10.0 (fabs a))))))))))

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 <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * fabs(a);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = k * ((-10.0 * fabs(a)) + (fabs(a) / k));
	} else {
		tmp = fabs(a) + (k * ((-1.0 * (k * (fabs(a) + (-100.0 * fabs(a))))) - (10.0 * fabs(a))));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = (Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * Math.abs(a);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = k * ((-10.0 * Math.abs(a)) + (Math.abs(a) / k));
	} else {
		tmp = Math.abs(a) + (k * ((-1.0 * (k * (Math.abs(a) + (-100.0 * Math.abs(a))))) - (10.0 * Math.abs(a))));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = (math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_0 <= 1e+298:
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * math.fabs(a)
	elif t_0 <= math.inf:
		tmp = k * ((-10.0 * math.fabs(a)) + (math.fabs(a) / k))
	else:
		tmp = math.fabs(a) + (k * ((-1.0 * (k * (math.fabs(a) + (-100.0 * math.fabs(a))))) - (10.0 * math.fabs(a))))
	return math.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 <= 1e+298)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(k * Float64(10.0 + k)))) * abs(a));
	elseif (t_0 <= Inf)
		tmp = Float64(k * Float64(Float64(-10.0 * abs(a)) + Float64(abs(a) / k)));
	else
		tmp = Float64(abs(a) + Float64(k * Float64(Float64(-1.0 * Float64(k * Float64(abs(a) + Float64(-100.0 * abs(a))))) - Float64(10.0 * abs(a)))));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_0 <= 1e+298)
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * abs(a);
	elseif (t_0 <= Inf)
		tmp = k * ((-10.0 * abs(a)) + (abs(a) / k));
	else
		tmp = abs(a) + (k * ((-1.0 * (k * (abs(a) + (-100.0 * abs(a))))) - (10.0 * abs(a))));
	end
	tmp_2 = (sign(a) * abs(1.0)) * 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, 1e+298], N[(N[(1.0 / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(k * N[(N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[a], $MachinePrecision] + N[(k * N[(N[(-1.0 * N[(k * N[(N[Abs[a], $MachinePrecision] + N[(-100.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999996e297

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot {k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

   3. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(k - -10\right) \cdot k - -1} \cdot a} \]

   4. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \cdot a \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \cdot a \]

      4. lower-+.f64 46.1%

         \[\leadsto \frac{1}{1 + k \cdot \left(10 + \color{blue}{k}\right)} \cdot a \]

   6. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

   if 9.9999999999999996e297 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      4. lower-/.f64 20.2%

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

   10. Applied rewrites 20.2%

       \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]

   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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

      3. associate-*r* N/A

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

      4. lower-*.f64 N/A

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

      5. lower-*.f64 21.5%

         \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

   9. Applied rewrites 21.5%

      \[\leadsto a + \left(-10 \cdot a\right) \cdot k \]

   10. Taylor expanded in k around 0

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto a + k \cdot \color{blue}{\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - \color{blue}{10 \cdot a}\right) \]

       3. lower--.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot \color{blue}{a}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       6. lower-+.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

       8. lower-*.f64 27.8%

          \[\leadsto a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   12. Applied rewrites 27.8%

       \[\leadsto a + \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.3% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+298}:\\ \;\;\;\;\frac{1}{1 + k \cdot \left(10 + k\right)} \cdot \left|a\right|\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right| + \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (*
 (copysign 1.0 a)
 (if (<=
      (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
      1e+298)
   (* (/ 1.0 (+ 1.0 (* k (+ 10.0 k)))) (fabs a))
   (* k (+ (* -10.0 (fabs a)) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * fabs(a);
	} else {
		tmp = k * ((-10.0 * fabs(a)) + (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (((Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298) {
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * Math.abs(a);
	} else {
		tmp = k * ((-10.0 * Math.abs(a)) + (Math.abs(a) / k));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if ((math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298:
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * math.fabs(a)
	else:
		tmp = k * ((-10.0 * math.fabs(a)) + (math.fabs(a) / k))
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+298)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(k * Float64(10.0 + k)))) * abs(a));
	else
		tmp = Float64(k * Float64(Float64(-10.0 * abs(a)) + Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (((abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298)
		tmp = (1.0 / (1.0 + (k * (10.0 + k)))) * abs(a);
	else
		tmp = k * ((-10.0 * abs(a)) + (abs(a) / k));
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+298], N[(N[(1.0 / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], N[(k * N[(N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999996e297

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot {k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

   3. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(k - -10\right) \cdot k - -1} \cdot a} \]

   4. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \cdot a \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \cdot a \]

      4. lower-+.f64 46.1%

         \[\leadsto \frac{1}{1 + k \cdot \left(10 + \color{blue}{k}\right)} \cdot a \]

   6. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

   if 9.9999999999999996e297 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      4. lower-/.f64 20.2%

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

   10. Applied rewrites 20.2%

       \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.3% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+298}:\\ \;\;\;\;\frac{\left|a\right|}{\left(k - -10\right) \cdot k - -1}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right| + \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (*
 (copysign 1.0 a)
 (if (<=
      (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
      1e+298)
   (/ (fabs a) (- (* (- k -10.0) k) -1.0))
   (* k (+ (* -10.0 (fabs a)) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298) {
		tmp = fabs(a) / (((k - -10.0) * k) - -1.0);
	} else {
		tmp = k * ((-10.0 * fabs(a)) + (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (((Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298) {
		tmp = Math.abs(a) / (((k - -10.0) * k) - -1.0);
	} else {
		tmp = k * ((-10.0 * Math.abs(a)) + (Math.abs(a) / k));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if ((math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298:
		tmp = math.fabs(a) / (((k - -10.0) * k) - -1.0)
	else:
		tmp = k * ((-10.0 * math.fabs(a)) + (math.fabs(a) / k))
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+298)
		tmp = Float64(abs(a) / Float64(Float64(Float64(k - -10.0) * k) - -1.0));
	else
		tmp = Float64(k * Float64(Float64(-10.0 * abs(a)) + Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (((abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+298)
		tmp = abs(a) / (((k - -10.0) * k) - -1.0);
	else
		tmp = k * ((-10.0 * abs(a)) + (abs(a) / k));
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+298], N[(N[Abs[a], $MachinePrecision] / N[(N[(N[(k - -10.0), $MachinePrecision] * k), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999996e297

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      3. lift-*.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{k \cdot k} \cdot \color{blue}{\sqrt{k \cdot k}}\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      7. lower-unsound-*.f32 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      9. lower-unsound-*.f64 46.1%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

   6. Applied rewrites 46.1%

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) + \color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) + \left(\mathsf{neg}\left(-1\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) - \color{blue}{-1}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) - -1} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      9. sqrt-unprod N/A

         \[\leadsto \frac{a}{\left(\sqrt{k \cdot k} \cdot \sqrt{k \cdot k} + 10 \cdot k\right) - -1} \]

      10. rem-square-sqrt N/A

          \[\leadsto \frac{a}{\left(k \cdot k + 10 \cdot k\right) - -1} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      14. distribute-rgt-in N/A

          \[\leadsto \frac{a}{k \cdot \left(10 + k\right) - -1} \]

      15. *-commutative N/A

          \[\leadsto \frac{a}{\left(10 + k\right) \cdot k - -1} \]

      16. +-commutative N/A

          \[\leadsto \frac{a}{\left(k + 10\right) \cdot k - -1} \]

      17. metadata-eval N/A

          \[\leadsto \frac{a}{\left(k + \left(\mathsf{neg}\left(-10\right)\right)\right) \cdot k - -1} \]

      18. sub-flip N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

      19. lift--.f64 N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

   8. Applied rewrites 46.1%

      \[\leadsto \frac{a}{\color{blue}{\left(k - -10\right) \cdot k - -1}} \]

   if 9.9999999999999996e297 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

      4. lower-/.f64 20.2%

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{k}\right) \]

   10. Applied rewrites 20.2%

       \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.2% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+183}:\\ \;\;\;\;\frac{\left|a\right|}{\left(k - -10\right) \cdot k - -1}\\ \mathbf{else}:\\ \;\;\;\;\left|a\right| + -10 \cdot \left(\left|a\right| \cdot k\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
  :precision binary64
  (*
 (copysign 1.0 a)
 (if (<=
      (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
      1e+183)
   (/ (fabs a) (- (* (- k -10.0) k) -1.0))
   (+ (fabs a) (* -10.0 (* (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+183) {
		tmp = fabs(a) / (((k - -10.0) * k) - -1.0);
	} else {
		tmp = fabs(a) + (-10.0 * (fabs(a) * k));
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (((Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+183) {
		tmp = Math.abs(a) / (((k - -10.0) * k) - -1.0);
	} else {
		tmp = Math.abs(a) + (-10.0 * (Math.abs(a) * k));
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if ((math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+183:
		tmp = math.fabs(a) / (((k - -10.0) * k) - -1.0)
	else:
		tmp = math.fabs(a) + (-10.0 * (math.fabs(a) * k))
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+183)
		tmp = Float64(abs(a) / Float64(Float64(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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (((abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+183)
		tmp = abs(a) / (((k - -10.0) * k) - -1.0);
	else
		tmp = abs(a) + (-10.0 * (abs(a) * k));
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e+183], N[(N[Abs[a], $MachinePrecision] / N[(N[(N[(k - -10.0), $MachinePrecision] * k), $MachinePrecision] - -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 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))) < 9.9999999999999995e182

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      3. lift-*.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      4. rem-square-sqrt N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{k \cdot k} \cdot \color{blue}{\sqrt{k \cdot k}}\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      7. lower-unsound-*.f32 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

      9. lower-unsound-*.f64 46.1%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]

   6. Applied rewrites 46.1%

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) + \color{blue}{1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) + \left(\mathsf{neg}\left(-1\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) - \color{blue}{-1}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{a}{\left(10 \cdot k + \sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right) - -1} \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{a}{\left(\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} + 10 \cdot k\right) - -1} \]

      9. sqrt-unprod N/A

         \[\leadsto \frac{a}{\left(\sqrt{k \cdot k} \cdot \sqrt{k \cdot k} + 10 \cdot k\right) - -1} \]

      10. rem-square-sqrt N/A

          \[\leadsto \frac{a}{\left(k \cdot k + 10 \cdot k\right) - -1} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(10 \cdot k + k \cdot k\right) - -1} \]

      14. distribute-rgt-in N/A

          \[\leadsto \frac{a}{k \cdot \left(10 + k\right) - -1} \]

      15. *-commutative N/A

          \[\leadsto \frac{a}{\left(10 + k\right) \cdot k - -1} \]

      16. +-commutative N/A

          \[\leadsto \frac{a}{\left(k + 10\right) \cdot k - -1} \]

      17. metadata-eval N/A

          \[\leadsto \frac{a}{\left(k + \left(\mathsf{neg}\left(-10\right)\right)\right) \cdot k - -1} \]

      18. sub-flip N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

      19. lift--.f64 N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{a}{\left(k - -10\right) \cdot k - -1} \]

   8. Applied rewrites 46.1%

      \[\leadsto \frac{a}{\color{blue}{\left(k - -10\right) \cdot k - -1}} \]

   if 9.9999999999999995e182 < (/.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 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

      5. lower-pow.f64 46.0%

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. 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 21.6%

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 21.6% accurate, 9.6× 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 90.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

2. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{a}{\color{blue}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{a}{1 + \color{blue}{\left(10 \cdot k + {k}^{2}\right)}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{{k}^{2}}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + {\color{blue}{k}}^{2}\right)} \]

   5. lower-pow.f64 46.0%

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + {k}^{\color{blue}{2}}\right)} \]

4. Applied rewrites 46.0%

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

5. Taylor expanded in k around 0

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
6. 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 21.6%

      \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

7. Applied rewrites 21.6%

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))