Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 99.7%

Time: 4.6s

Alternatives: 13

Speedup: 1.3×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{k - -10}{t\_0}, k, \frac{{k}^{\left(-m\right)}}{a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 1e-61)
     t_0
     (/ 1.0 (fma (/ (- k -10.0) t_0) k (/ (pow k (- m)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 1e-61) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(((k - -10.0) / t_0), k, (pow(k, -m) / a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 1e-61)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(Float64(Float64(k - -10.0) / t_0), k, Float64((k ^ Float64(-m)) / a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 1e-61], t$95$0, N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] / t$95$0), $MachinePrecision] * k + N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1e-61

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 83.6

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

   6. Applied rewrites 83.6%

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

   if 1e-61 < k

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.4

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

      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. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. +-lft-identity N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. add-flip N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 90.5

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 90.5

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

   3. Applied rewrites 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow-flip N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-neg.f64 88.3

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

   5. Applied rewrites 88.3%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-110}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{elif}\;m \leq 6.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6e-110)
   (* (/ (pow k m) (fma (- k -10.0) k 1.0)) a)
   (if (<= m 6.3e-25)
     (/ 1.0 (fma (- k -10.0) (/ k a) (/ 1.0 a)))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6e-110) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * a;
	} else if (m <= 6.3e-25) {
		tmp = 1.0 / fma((k - -10.0), (k / a), (1.0 / a));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6e-110)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * a);
	elseif (m <= 6.3e-25)
		tmp = Float64(1.0 / fma(Float64(k - -10.0), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6e-110], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 6.3e-25], N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.99999999999999972e-110

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.6%

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

   if -5.99999999999999972e-110 < m < 6.29999999999999961e-25

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 44.4

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

   8. Applied rewrites 44.4%

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

   if 6.29999999999999961e-25 < m

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 83.6

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

   6. Applied rewrites 83.6%

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

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -0.0017:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 6.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -0.0017)
     t_0
     (if (<= m 6.3e-25) (/ 1.0 (fma (- k -10.0) (/ k a) (/ 1.0 a))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -0.0017) {
		tmp = t_0;
	} else if (m <= 6.3e-25) {
		tmp = 1.0 / fma((k - -10.0), (k / a), (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -0.0017)
		tmp = t_0;
	elseif (m <= 6.3e-25)
		tmp = Float64(1.0 / fma(Float64(k - -10.0), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -0.0017], t$95$0, If[LessEqual[m, 6.3e-25], N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -0.00169999999999999991 or 6.29999999999999961e-25 < m

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 83.6

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

   6. Applied rewrites 83.6%

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

   if -0.00169999999999999991 < m < 6.29999999999999961e-25

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 44.4

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

   8. Applied rewrites 44.4%

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

Alternative 4: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+223}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{\frac{a}{10 + k}}, k, \frac{{k}^{\left(-m\right)}}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+223)
   (/ 1.0 (fma (/ 1.0 (/ a (+ 10.0 k))) k (/ (pow k (- m)) a)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+223) {
		tmp = 1.0 / fma((1.0 / (a / (10.0 + k))), k, (pow(k, -m) / a));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+223)
		tmp = Float64(1.0 / fma(Float64(1.0 / Float64(a / Float64(10.0 + k))), k, Float64((k ^ Float64(-m)) / a)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+223], N[(1.0 / N[(N[(1.0 / N[(a / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e223

   1. Initial program 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.4

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

      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. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. +-lft-identity N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. add-flip N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 90.5

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 90.5

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

   3. Applied rewrites 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow-flip N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-neg.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 71.9

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

   10. Applied rewrites 71.9%

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

   if 1.00000000000000005e223 < (/.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 90.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 90.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 83.6

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

   6. Applied rewrites 83.6%

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

Alternative 5: 55.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{a}}\\ \mathbf{elif}\;m \leq 6.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5e+45)
   (/ 1.0 (/ (fma (- k -10.0) k 1.0) a))
   (if (<= m 6.3e-25)
     (/ 1.0 (fma (- k -10.0) (/ k a) (/ 1.0 a)))
     (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5e+45) {
		tmp = 1.0 / (fma((k - -10.0), k, 1.0) / a);
	} else if (m <= 6.3e-25) {
		tmp = 1.0 / fma((k - -10.0), (k / a), (1.0 / a));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5e+45)
		tmp = Float64(1.0 / Float64(fma(Float64(k - -10.0), k, 1.0) / a));
	elseif (m <= 6.3e-25)
		tmp = Float64(1.0 / fma(Float64(k - -10.0), Float64(k / a), Float64(1.0 / a)));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5e+45], N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.3e-25], N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5e45

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if -5e45 < m < 6.29999999999999961e-25

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 44.4

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

   8. Applied rewrites 44.4%

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

   if 6.29999999999999961e-25 < m

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2

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

   9. Applied rewrites 29.2%

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

Alternative 6: 52.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+223}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+223)
   (/ 1.0 (/ (fma (- k -10.0) k 1.0) a))
   (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+223) {
		tmp = 1.0 / (fma((k - -10.0), k, 1.0) / a);
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+223)
		tmp = Float64(1.0 / Float64(fma(Float64(k - -10.0), k, 1.0) / a));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+223], N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e223

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if 1.00000000000000005e223 < (/.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 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2

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

   9. Applied rewrites 29.2%

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

Alternative 7: 52.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+223}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+223)
   (/ a (fma (- k -10.0) k 1.0))
   (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+223) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+223)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+223], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000000000000005e223

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if 1.00000000000000005e223 < (/.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 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2

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

   9. Applied rewrites 29.2%

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

Alternative 8: 47.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+275}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+275)
   (/ a (fma (- k -10.0) k 1.0))
   (* k (fma -10.0 a (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+275) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * fma(-10.0, a, (a / k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+275)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * fma(-10.0, a, Float64(a / k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+275], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * a + N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999996e274

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if 9.9999999999999996e274 < (/.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 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

   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 21.2

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

   7. Applied rewrites 21.2%

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

   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-fma.f64 N/A

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

      3. lower-/.f64 19.9

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

   10. Applied rewrites 19.9%

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

Alternative 9: 46.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 6.8e+15) (/ a (fma (- k -10.0) k 1.0)) (fma (* a k) -10.0 a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.8e+15) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma((a * k), -10.0, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6.8e+15)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = fma(Float64(a * k), -10.0, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6.8e+15], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8e15

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if 6.8e15 < m

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

   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 21.2

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

   7. Applied rewrites 21.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 21.2

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

   9. Applied rewrites 21.2%

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

Alternative 10: 30.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 6.8e+15) (/ a (fma 10.0 k 1.0)) (fma (* a k) -10.0 a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.8e+15) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = fma((a * k), -10.0, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6.8e+15)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = fma(Float64(a * k), -10.0, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6.8e+15], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8e15

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.5

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 28.4%

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

   if 6.8e15 < m

   1. Initial program 90.6%

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

   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-fma.f64 N/A

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

      4. lower-pow.f64 44.5

         \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

   4. Applied rewrites 44.5%

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

   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 21.2

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

   7. Applied rewrites 21.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 21.2

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

   9. Applied rewrites 21.2%

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

Alternative 11: 21.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

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-fma.f64 N/A

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

   4. lower-pow.f64 44.5

      \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

4. Applied rewrites 44.5%

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

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 21.2

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

7. Applied rewrites 21.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 21.2

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

9. Applied rewrites 21.2%

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

Alternative 12: 21.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

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-fma.f64 N/A

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

   4. lower-pow.f64 44.5

      \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

4. Applied rewrites 44.5%

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

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 21.2

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

7. Applied rewrites 21.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

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

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

   9. lower-fma.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 21.2

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

9. Applied rewrites 21.2%

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

Alternative 13: 20.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (/ a 1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

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-fma.f64 N/A

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

   4. lower-pow.f64 44.5

      \[\leadsto \frac{a}{1 + \mathsf{fma}\left(10, k, {k}^{2}\right)} \]

4. Applied rewrites 44.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-flip N/A

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

   8. lift--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 44.5

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

6. Applied rewrites 44.5%

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

7. Taylor expanded in k around 0

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

9. Applied rewrites 20.3%

   \[\leadsto \frac{a}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))