Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.7%

Time: 4.5s

Alternatives: 13

Speedup: 1.3×

• 77.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{-1 \cdot \left(m \cdot \log k\right)}\\ t_1 := \frac{t\_0}{a}\\ \mathbf{if}\;k \leq 10^{-41}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k, \mathsf{fma}\left(10, t\_1, \frac{k \cdot t\_0}{a}\right), t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (exp (* -1.0 (* m (log k))))) (t_1 (/ t_0 a)))
   (if (<= k 1e-41)
     (* (pow k m) a)
     (/ 1.0 (fma k (fma 10.0 t_1 (/ (* k t_0) a)) t_1)))))

C

double code(double a, double k, double m) {
	double t_0 = exp((-1.0 * (m * log(k))));
	double t_1 = t_0 / a;
	double tmp;
	if (k <= 1e-41) {
		tmp = pow(k, m) * a;
	} else {
		tmp = 1.0 / fma(k, fma(10.0, t_1, ((k * t_0) / a)), t_1);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = exp(Float64(-1.0 * Float64(m * log(k))))
	t_1 = Float64(t_0 / a)
	tmp = 0.0
	if (k <= 1e-41)
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(1.0 / fma(k, fma(10.0, t_1, Float64(Float64(k * t_0) / a)), t_1));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Exp[N[(-1.0 * N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / a), $MachinePrecision]}, If[LessEqual[k, 1e-41], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(1.0 / N[(k * N[(10.0 * t$95$1 + N[(N[(k * t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000001e-41

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

   if 1.00000000000000001e-41 < k

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 82.3%

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 71.0%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|a\right|}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k, \mathsf{fma}\left(10, t\_0, \frac{k}{\left|a\right|}\right), t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs a))))
   (*
    (copysign 1.0 a)
    (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 4e+233)
      (/ (pow k m) (fma k (fma 10.0 t_0 (/ k (fabs a))) t_0))
      (* (pow k m) (fabs a))))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 / fabs(a);
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 4e+233) {
		tmp = pow(k, m) / fma(k, fma(10.0, t_0, (k / fabs(a))), t_0);
	} else {
		tmp = pow(k, m) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(1.0 / abs(a))
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 4e+233)
		tmp = Float64((k ^ m) / fma(k, fma(10.0, t_0, Float64(k / abs(a))), t_0));
	else
		tmp = Float64((k ^ m) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 / N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+233], N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(10.0 * t$95$0 + N[(k / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999989e233

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 82.3%

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 91.8%

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

   11. Applied rewrites 91.8%

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

   if 3.99999999999999989e233 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} \cdot 1}\\ \mathbf{elif}\;m \leq 0.41:\\ \;\;\;\;\frac{\left(1 + m \cdot \log k\right) \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.85e-6)
   (/ 1.0 (* (/ (pow k (- m)) a) 1.0))
   (if (<= m 0.41)
     (/ (* (+ 1.0 (* m (log k))) a) (fma (- k -10.0) k 1.0))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.85e-6) {
		tmp = 1.0 / ((pow(k, -m) / a) * 1.0);
	} else if (m <= 0.41) {
		tmp = ((1.0 + (m * log(k))) * a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.85e-6)
		tmp = Float64(1.0 / Float64(Float64((k ^ Float64(-m)) / a) * 1.0));
	elseif (m <= 0.41)
		tmp = Float64(Float64(Float64(1.0 + Float64(m * log(k))) * a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.85e-6], N[(1.0 / N[(N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.41], N[(N[(N[(1.0 + N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.8500000000000001e-6

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 82.3%

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

   6. Applied rewrites 82.3%

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

   if -1.8500000000000001e-6 < m < 0.409999999999999976

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.5%

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

   6. Applied rewrites 41.5%

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

   if 0.409999999999999976 < m

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.41:\\ \;\;\;\;\frac{\left(1 + m \cdot \log k\right) \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.85e-6)
     t_0
     (if (<= m 0.41)
       (/ (* (+ 1.0 (* m (log k))) a) (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 <= -1.85e-6) {
		tmp = t_0;
	} else if (m <= 0.41) {
		tmp = ((1.0 + (m * log(k))) * a) / 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 <= -1.85e-6)
		tmp = t_0;
	elseif (m <= 0.41)
		tmp = Float64(Float64(Float64(1.0 + Float64(m * log(k))) * a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.85e-6], t$95$0, If[LessEqual[m, 0.41], N[(N[(N[(1.0 + N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.8500000000000001e-6 or 0.409999999999999976 < m

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

   if -1.8500000000000001e-6 < m < 0.409999999999999976

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.5%

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

   6. Applied rewrites 41.5%

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

Alternative 5: 97.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

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

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.5%

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

   9. Applied rewrites 29.5%

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

Alternative 6: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 1.36:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.36) (* (/ (pow k m) (fma (- k -10.0) k 1.0)) a) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.36) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.36)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.36], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3600000000000001

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   if 1.3600000000000001 < m

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

Alternative 7: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 53:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.85e-6)
     t_0
     (if (<= m 53.0) (/ a (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 <= -1.85e-6) {
		tmp = t_0;
	} else if (m <= 53.0) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.85e-6)
		tmp = t_0;
	elseif (m <= 53.0)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.85e-6], t$95$0, If[LessEqual[m, 53.0], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.8500000000000001e-6 or 53 < m

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.4%

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

   9. Applied rewrites 82.4%

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

   if -1.8500000000000001e-6 < m < 53

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

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

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

      14. distribute-rgt-neg-out N/A

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

      15. sub-negate-rev N/A

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

      16. sub-flip N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 45.9%

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. metadata-eval N/A

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

      25. sub-flip N/A

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

      26. lower--.f64 45.9%

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

   8. Applied rewrites 45.9%

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

Alternative 8: 55.3% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 4e+233)
    (/ (fabs a) (fma (- k -10.0) k 1.0))
    (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 4e+233) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 4e+233)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+233], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.99999999999999989e233

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

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

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

      14. distribute-rgt-neg-out N/A

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

      15. sub-negate-rev N/A

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

      16. sub-flip N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 45.9%

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. metadata-eval N/A

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

      25. sub-flip N/A

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

      26. lower--.f64 45.9%

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

   8. Applied rewrites 45.9%

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

   if 3.99999999999999989e233 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.5%

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

   9. Applied rewrites 29.5%

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

Alternative 9: 50.5% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 5e+303)
    (/ (fabs a) (fma (- k -10.0) k 1.0))
    (* k (fma -10.0 (fabs a) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 5e+303) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 5e+303)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+303], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.9999999999999997e303

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

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

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

      14. distribute-rgt-neg-out N/A

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

      15. sub-negate-rev N/A

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

      16. sub-flip N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 45.9%

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. metadata-eval N/A

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

      25. sub-flip N/A

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

      26. lower--.f64 45.9%

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

   8. Applied rewrites 45.9%

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

   if 4.9999999999999997e303 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

   10. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 20.1%

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

   12. Applied rewrites 20.1%

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

Alternative 10: 47.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-flip N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

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

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

      14. distribute-rgt-neg-out N/A

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

      15. sub-negate-rev N/A

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

      16. sub-flip N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 45.9%

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. metadata-eval N/A

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

      25. sub-flip N/A

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

      26. lower--.f64 45.9%

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

   8. Applied rewrites 45.9%

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

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

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

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

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

       9. lower-fma.f64 N/A

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

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

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

       11. metadata-eval N/A

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

       12. lower-*.f64 21.5%

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

   11. Applied rewrites 21.5%

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

Alternative 11: 30.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 29.1%

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

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

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-lft-identity N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 90.5%

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5%

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

   9. Applied rewrites 21.5%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

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

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

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

       9. lower-fma.f64 N/A

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

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

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

       11. metadata-eval N/A

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

       12. lower-*.f64 21.5%

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

   11. Applied rewrites 21.5%

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

Alternative 12: 21.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 90.4%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. +-lft-identity N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-rgt-out N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. +-commutative N/A

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

   20. add-flip N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 90.5%

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

3. Applied rewrites 90.5%

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

4. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 45.9%

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

6. Applied rewrites 45.9%

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

7. Taylor expanded in k around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 21.5%

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

9. Applied rewrites 21.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. metadata-eval N/A

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

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

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

    9. lower-fma.f64 N/A

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

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

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(10\right)\right) \cdot k, a, a\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]

    12. lower-*.f64 21.5%

        \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]

11. Applied rewrites 21.5%

    \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]
12. Add Preprocessing

Alternative 13: 20.7% accurate, 8.7× speedup

Math

\[1 \cdot a \]

FPCore

(FPCore (a k m) :precision binary64 (* 1.0 a))

C

double code(double a, double k, double m) {
	return 1.0 * a;
}

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 = 1.0d0 * a
end function

Java

public static double code(double a, double k, double m) {
	return 1.0 * a;
}

Python

def code(a, k, m):
	return 1.0 * a

Julia

function code(a, k, m)
	return Float64(1.0 * a)
end

MATLAB

function tmp = code(a, k, m)
	tmp = 1.0 * a;
end

Wolfram

code[a_, k_, m_] := N[(1.0 * a), $MachinePrecision]

Derivation

1. Initial program 90.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{a \cdot {k}^{m}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot a} \]

3. Applied rewrites 90.4%

   \[\leadsto \color{blue}{\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a} \]

4. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \cdot a \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{1 + k \cdot \color{blue}{\left(10 + k\right)}} \cdot a \]

   4. lower-+.f64 45.9%

      \[\leadsto \frac{1}{1 + k \cdot \left(10 + \color{blue}{k}\right)} \cdot a \]

6. Applied rewrites 45.9%

   \[\leadsto \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \cdot a \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{1}{1 + k \cdot 10} \cdot a \]
8. Step-by-step derivation

9. Applied rewrites 29.1%

   \[\leadsto \frac{1}{1 + k \cdot 10} \cdot a \]

10. Taylor expanded in k around 0

    \[\leadsto 1 \cdot a \]
11. Step-by-step derivation

12. Applied rewrites 20.7%

    \[\leadsto 1 \cdot a \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025179 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))