Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 99.6%

Time: 5.0s

Alternatives: 13

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{10}{t\_0} + \frac{k}{t\_0}, k, \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 5e-25)
     t_0
     (/ 1.0 (fma (+ (/ 10.0 t_0) (/ k t_0)) k (/ 1.0 t_0))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 5e-25) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(((10.0 / t_0) + (k / t_0)), k, (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 5e-25)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(Float64(Float64(10.0 / t_0) + Float64(k / t_0)), k, Float64(1.0 / 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[k, 5e-25], t$95$0, N[(1.0 / N[(N[(N[(10.0 / t$95$0), $MachinePrecision] + N[(k / t$95$0), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999962e-25

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if 4.99999999999999962e-25 < k

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-pow.f64 N/A

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

   6. Applied rewrites 82.6%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 1.5 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{k - -10}{t\_0}, k, \frac{{k}^{\left(-m\right)}}{a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 1.5e-41)
     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 <= 1.5e-41) {
		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 <= 1.5e-41)
		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, 1.5e-41], 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 < 1.49999999999999994e-41

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if 1.49999999999999994e-41 < k

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   10. Applied rewrites 88.4%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{elif}\;m \leq 0.008:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k - -10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.4e-9)
   (/ 1.0 (/ (pow k (- m)) a))
   (if (<= m 0.008)
     (/ 1.0 (fma k (/ (- k -10.0) a) (/ 1.0 a)))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.4e-9) {
		tmp = 1.0 / (pow(k, -m) / a);
	} else if (m <= 0.008) {
		tmp = 1.0 / fma(k, ((k - -10.0) / a), (1.0 / a));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.4e-9)
		tmp = Float64(1.0 / Float64((k ^ Float64(-m)) / a));
	elseif (m <= 0.008)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k - -10.0) / a), Float64(1.0 / a)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.4e-9], N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.008], N[(1.0 / N[(k * N[(N[(k - -10.0), $MachinePrecision] / 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 < -7.4e-9

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if -7.4e-9 < m < 0.0080000000000000002

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 44.9

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

   8. Applied rewrites 44.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{a}}} \]
   9. 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-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 44.7

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

   10. Applied rewrites 44.7%

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

   if 0.0080000000000000002 < m

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 82.9

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

   8. Applied rewrites 82.9%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.008:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k - -10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -7.4e-9)
     t_0
     (if (<= m 0.008) (/ 1.0 (fma k (/ (- k -10.0) 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 <= -7.4e-9) {
		tmp = t_0;
	} else if (m <= 0.008) {
		tmp = 1.0 / fma(k, ((k - -10.0) / 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 <= -7.4e-9)
		tmp = t_0;
	elseif (m <= 0.008)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k - -10.0) / 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, -7.4e-9], t$95$0, If[LessEqual[m, 0.008], N[(1.0 / N[(k * N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -7.4e-9 or 0.0080000000000000002 < m

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if -7.4e-9 < m < 0.0080000000000000002

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 44.9

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

   8. Applied rewrites 44.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{a}}} \]
   9. 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-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 44.7

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

   10. Applied rewrites 44.7%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 5.5:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 5.5) (/ t_0 (fma (- k -10.0) k 1.0)) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 5.5) {
		tmp = t_0 / fma((k - -10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 5.5)
		tmp = Float64(t_0 / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, 5.5], N[(t$95$0 / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-special-/ N/A

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

       2. associate-/l/ N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-special-/ N/A

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

       7. distribute-rgt-in N/A

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

       8. pow2 N/A

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

       9. associate-+l+ N/A

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

       10. pow2 N/A

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

       11. division-flip N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-pow.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. +-commutative N/A

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

   11. Applied rewrites 90.5%

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

   if 5.5 < m

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 82.9

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

   8. Applied rewrites 82.9%

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

Alternative 6: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.054:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \frac{k - -10}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.054)
   (/ a (* k k))
   (if (<= m 1.9)
     (/ 1.0 (fma k (/ (- k -10.0) a) (/ 1.0 a)))
     (fma (- (- (* (* -99.0 a) k)) (* 10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.054) {
		tmp = a / (k * k);
	} else if (m <= 1.9) {
		tmp = 1.0 / fma(k, ((k - -10.0) / a), (1.0 / a));
	} else {
		tmp = fma((-((-99.0 * a) * k) - (10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.054)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.9)
		tmp = Float64(1.0 / fma(k, Float64(Float64(k - -10.0) / a), Float64(1.0 / a)));
	else
		tmp = fma(Float64(Float64(-Float64(Float64(-99.0 * a) * k)) - Float64(10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.054], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9], N[(1.0 / N[(k * N[(N[(k - -10.0), $MachinePrecision] / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(-99.0 * a), $MachinePrecision] * k), $MachinePrecision]) - N[(10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0539999999999999994

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   if -0.0539999999999999994 < m < 1.8999999999999999

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-/.f64 N/A

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

      10. division-flip N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative 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. lower-+.f64 N/A

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

      21. *-commutative N/A

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

   3. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-pow.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 44.9

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

   8. Applied rewrites 44.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(k + 10, k, 1\right)}{a}}} \]
   9. 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-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 44.7

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

   10. Applied rewrites 44.7%

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

   if 1.8999999999999999 < m

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right) - 10 \cdot a, k, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      12. lower-*.f64 27.4

          \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

   7. Applied rewrites 27.4%

      \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, \color{blue}{k}, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.054:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.9:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.054)
   (/ a (* k k))
   (if (<= m 1.9)
     (/ a (fma (- k -10.0) k 1.0))
     (fma (- (- (* (* -99.0 a) k)) (* 10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.054) {
		tmp = a / (k * k);
	} else if (m <= 1.9) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma((-((-99.0 * a) * k) - (10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.054)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.9)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = fma(Float64(Float64(-Float64(Float64(-99.0 * a) * k)) - Float64(10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.054], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.9], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(-99.0 * a), $MachinePrecision] * k), $MachinePrecision]) - N[(10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0539999999999999994

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   if -0.0539999999999999994 < m < 1.8999999999999999

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in m around 0

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

      1. pow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 45.0

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

   10. Applied rewrites 45.0%

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

   if 1.8999999999999999 < m

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right) - 10 \cdot a, k, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      12. lower-*.f64 27.4

          \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

   7. Applied rewrites 27.4%

      \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, \color{blue}{k}, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.054:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{+133}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log k, m, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.054)
   (/ a (* k k))
   (if (<= m 2.9e+133)
     (/ a (fma (- k -10.0) k 1.0))
     (* (fma (log k) m 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.054) {
		tmp = a / (k * k);
	} else if (m <= 2.9e+133) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma(log(k), m, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.054)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.9e+133)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(fma(log(k), m, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.054], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.9e+133], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[k], $MachinePrecision] * m + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0539999999999999994

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   if -0.0539999999999999994 < m < 2.9000000000000001e133

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in m around 0

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

      1. pow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 45.0

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

   10. Applied rewrites 45.0%

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

   if 2.9000000000000001e133 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 40.8

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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

         \[\leadsto a + \left(m \cdot \log k\right) \cdot a \]

      2. *-commutative N/A

         \[\leadsto a + \left(\log k \cdot m\right) \cdot a \]

      3. +-commutative N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a + a \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\log k \cdot m + 1\right) \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\log k \cdot m + 1\right) \cdot a \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\log k \cdot m + 1\right) \cdot a \]

      7. lift-fma.f64 23.4

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

   7. Applied rewrites 23.4%

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

Alternative 9: 48.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+307)
   (/ a (fma (- k -10.0) k 1.0))
   (/ a (* k k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+307) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+307)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $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))) < 1.99999999999999997e307

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in m around 0

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

      1. pow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 45.0

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

   10. Applied rewrites 45.0%

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

   if 1.99999999999999997e307 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.5%

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

   2. Taylor expanded in 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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

Alternative 10: 47.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 2e-309) t_0 (if (<= k 10.0) (/ a (fma 10.0 k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2e-309) {
		tmp = t_0;
	} else if (k <= 10.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 2e-309)
		tmp = t_0;
	elseif (k <= 10.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-309], t$95$0, If[LessEqual[k, 10.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999988e-309 or 10 < k

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   if 1.9999999999999988e-309 < k < 10

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 28.2%

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

Alternative 11: 46.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 2e-309) t_0 (if (<= k 0.1) (fma (* a k) -10.0 a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2e-309) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = fma((a * k), -10.0, a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 2e-309)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = fma(Float64(a * k), -10.0, a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-309], t$95$0, If[LessEqual[k, 0.1], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999988e-309 or 0.10000000000000001 < k

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

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

   if 1.9999999999999988e-309 < k < 0.10000000000000001

   1. Initial program 90.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 20.9

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

   7. Applied rewrites 20.9%

      \[\leadsto \mathsf{fma}\left(a \cdot k, \color{blue}{-10}, a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1.6:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 2e-309) t_0 (if (<= k 1.6) (/ a 1.0) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2e-309) {
		tmp = t_0;
	} else if (k <= 1.6) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 2d-309) then
        tmp = t_0
    else if (k <= 1.6d0) then
        tmp = a / 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2e-309) {
		tmp = t_0;
	} else if (k <= 1.6) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 2e-309:
		tmp = t_0
	elif k <= 1.6:
		tmp = a / 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 2e-309)
		tmp = t_0;
	elseif (k <= 1.6)
		tmp = Float64(a / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 2e-309)
		tmp = t_0;
	elseif (k <= 1.6)
		tmp = a / 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-309], t$95$0, If[LessEqual[k, 1.6], N[(a / 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999988e-309 or 1.6000000000000001 < k

   1. Initial program 90.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   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. pow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \left(10 + k\right) + \color{blue}{1}} \]

      5. *-commutative N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, \color{blue}{k}, 1\right)} \]

      7. lower-+.f64 45.0

         \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)} \]

   4. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a}{{k}^{\color{blue}{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{a}{k \cdot k} \]

      3. lift-*.f64 35.5

         \[\leadsto \frac{a}{k \cdot k} \]

   7. Applied rewrites 35.5%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if 1.9999999999999988e-309 < k < 1.6000000000000001

   1. Initial program 90.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   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. pow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{k \cdot \left(10 + k\right) + \color{blue}{1}} \]

      5. *-commutative N/A

         \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, \color{blue}{k}, 1\right)} \]

      7. lower-+.f64 45.0

         \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)} \]

   4. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{a}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.0%

      \[\leadsto \frac{a}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 20.0% 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 90.5%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

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. pow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + k \cdot \color{blue}{k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{k \cdot \left(10 + k\right) + \color{blue}{1}} \]

   5. *-commutative N/A

      \[\leadsto \frac{a}{\left(10 + k\right) \cdot k + 1} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, \color{blue}{k}, 1\right)} \]

   7. lower-+.f64 45.0

      \[\leadsto \frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)} \]

4. Applied rewrites 45.0%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}} \]

5. Taylor expanded in k around 0

   \[\leadsto \frac{a}{1} \]
6. Step-by-step derivation

7. Applied rewrites 20.0%

   \[\leadsto \frac{a}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))