Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 97.7%

Time: 9.3s

Alternatives: 15

Speedup: 1.1×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.62) (* (pow k m) (/ 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 <= 1.62) {
		tmp = pow(k, m) * (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 <= 1.62)
		tmp = Float64((k ^ m) * Float64(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, 1.62], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 96.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 1.6200000000000001 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(1 - \frac{10 - {k}^{-1}}{k}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a - \frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* (* (- 1.0 (/ (- 10.0 (pow k -1.0)) k)) k) k))
     (if (<= t_0 5e+302)
       (/ a (fma (+ 10.0 k) k 1.0))
       (if (<= t_0 INFINITY)
         (/ (- a (/ (fma -99.0 (/ a k) (* 10.0 a)) k)) (* k k))
         (fma (* (- (* 99.0 k) 10.0) k) a a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (((1.0 - ((10.0 - pow(k, -1.0)) / k)) * k) * k);
	} else if (t_0 <= 5e+302) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (a - (fma(-99.0, (a / k), (10.0 * a)) / k)) / (k * k);
	} else {
		tmp = fma((((99.0 * k) - 10.0) * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(Float64(Float64(1.0 - Float64(Float64(10.0 - (k ^ -1.0)) / k)) * k) * k));
	elseif (t_0 <= 5e+302)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(a - Float64(fma(-99.0, Float64(a / k), Float64(10.0 * a)) / k)) / Float64(k * k));
	else
		tmp = fma(Float64(Float64(Float64(99.0 * k) - 10.0) * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(1.0 - N[(N[(10.0 - N[Power[k, -1.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(a - N[(N[(-99.0 * N[(a / k), $MachinePrecision] + N[(10.0 * a), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 34.5%

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

   9. Applied rewrites 42.5%

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

   10. Taylor expanded in k around -inf

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

   12. Applied rewrites 43.4%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5e302

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5e302 < (/.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 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in k around inf

      \[\leadsto \frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{\color{blue}{{k}^{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.7%

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

   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 0.0%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(1 - \frac{10 - {k}^{-1}}{k}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a - \frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(1 - \frac{10 - {k}^{-1}}{k}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* (* (- 1.0 (/ (- 10.0 (pow k -1.0)) k)) k) k))
     (if (<= t_0 5e+302)
       (/ a (fma (+ 10.0 k) k 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma (* (- (* 99.0 k) 10.0) k) a a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (((1.0 - ((10.0 - pow(k, -1.0)) / k)) * k) * k);
	} else if (t_0 <= 5e+302) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma((((99.0 * k) - 10.0) * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(Float64(Float64(1.0 - Float64(Float64(10.0 - (k ^ -1.0)) / k)) * k) * k));
	elseif (t_0 <= 5e+302)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(Float64(Float64(99.0 * k) - 10.0) * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(1.0 - N[(N[(10.0 - N[Power[k, -1.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 34.5%

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

   9. Applied rewrites 42.5%

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

   10. Taylor expanded in k around -inf

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

   12. Applied rewrites 43.4%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5e302

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5e302 < (/.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 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 43.2%

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

   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 0.0%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(1 - \frac{10 - {k}^{-1}}{k}\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(\frac{{k}^{-1} + 10}{k} - -1\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* (* (- (/ (+ (pow k -1.0) 10.0) k) -1.0) k) k))
     (if (<= t_0 5e+302)
       (/ a (fma (+ 10.0 k) k 1.0))
       (if (<= t_0 INFINITY)
         (/ a (* k k))
         (fma (* (- (* 99.0 k) 10.0) k) a a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (((((pow(k, -1.0) + 10.0) / k) - -1.0) * k) * k);
	} else if (t_0 <= 5e+302) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = a / (k * k);
	} else {
		tmp = fma((((99.0 * k) - 10.0) * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(Float64(Float64(Float64(Float64((k ^ -1.0) + 10.0) / k) - -1.0) * k) * k));
	elseif (t_0 <= 5e+302)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(Float64(Float64(99.0 * k) - 10.0) * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] + 10.0), $MachinePrecision] / k), $MachinePrecision] - -1.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 43.7%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5e302

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5e302 < (/.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 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 43.2%

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

   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 0.0%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\left(\left(\frac{{k}^{-1} + 10}{k} - -1\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 16.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
   (* (* a k) -10.0)
   (fma (* -10.0 a) k a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 0.0) {
		tmp = (a * k) * -10.0;
	} else {
		tmp = fma((-10.0 * a), k, 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))) <= 0.0)
		tmp = Float64(Float64(a * k) * -10.0);
	else
		tmp = fma(Float64(-10.0 * a), k, 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], 0.0], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k + 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))) < 0.0

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 17.2%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 6.9%

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

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

   1. Initial program 77.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 31.7%

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

   13. Applied rewrites 31.7%

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

Alternative 6: 16.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
   (* (* a k) -10.0)
   (* (fma -10.0 k 1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 0.0) {
		tmp = (a * k) * -10.0;
	} else {
		tmp = fma(-10.0, k, 1.0) * 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))) <= 0.0)
		tmp = Float64(Float64(a * k) * -10.0);
	else
		tmp = Float64(fma(-10.0, k, 1.0) * 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], 0.0], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $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))) < 0.0

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 17.2%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 6.9%

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

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

   1. Initial program 77.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 31.7%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 31.7%

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

Alternative 7: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.8e-8)
   (* (pow k m) (/ a (* k k)))
   (if (<= m 5.2e-7) (/ a (fma (+ 10.0 k) k 1.0)) (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.8e-8) {
		tmp = pow(k, m) * (a / (k * k));
	} else if (m <= 5.2e-7) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.8e-8)
		tmp = Float64((k ^ m) * Float64(a / Float64(k * k)));
	elseif (m <= 5.2e-7)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.8e-8], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.2e-7], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.7999999999999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -2.7999999999999999e-8 < m < 5.19999999999999998e-7

   1. Initial program 92.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 5.19999999999999998e-7 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -11000000 \lor \neg \left(m \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -11000000.0) (not (<= m 5.2e-7)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -11000000.0) || !(m <= 5.2e-7)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -11000000.0) || !(m <= 5.2e-7))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -11000000.0], N[Not[LessEqual[m, 5.2e-7]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.1e7 or 5.19999999999999998e-7 < m

   1. Initial program 89.2%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -1.1e7 < m < 5.19999999999999998e-7

   1. Initial program 93.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11000000 \lor \neg \left(m \leq 5.2 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.25:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+18)
   (/ a (* k k))
   (if (<= m 2.25)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (* (- (* 99.0 k) 10.0) k) a a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+18) {
		tmp = a / (k * k);
	} else if (m <= 2.25) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((((99.0 * k) - 10.0) * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.25)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(99.0 * k) - 10.0) * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.25], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e18

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 55.4%

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

   if -1.05e18 < m < 2.25

   1. Initial program 93.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 2.25 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 29.7%

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

Alternative 10: 61.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.25:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(99 \cdot k - 10\right) \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+18)
   (/ a (* k k))
   (if (<= m 2.25) (/ a (fma k k 1.0)) (fma (* (- (* 99.0 k) 10.0) k) a a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+18) {
		tmp = a / (k * k);
	} else if (m <= 2.25) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = fma((((99.0 * k) - 10.0) * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.25)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(99.0 * k) - 10.0) * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.25], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision] * k), $MachinePrecision] * a + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e18

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 55.4%

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

   if -1.05e18 < m < 2.25

   1. Initial program 93.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 90.3

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(-1 \cdot \left(k \cdot {\left(\sqrt{-1}\right)}^{2}\right), k, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 88.8%

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

   if 2.25 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 29.7%

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

Alternative 11: 56.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+18)
   (/ a (* k k))
   (if (<= m 1.15e+39) (/ a (fma k k 1.0)) (* (* a k) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+18) {
		tmp = a / (k * k);
	} else if (m <= 1.15e+39) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.15e+39)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(a * k) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15e+39], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e18

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 55.4%

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

   if -1.05e18 < m < 1.15000000000000006e39

   1. Initial program 90.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(-1 \cdot \left(k \cdot {\left(\sqrt{-1}\right)}^{2}\right), k, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 82.9%

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

   if 1.15000000000000006e39 < m

   1. Initial program 80.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 9.6%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 17.9%

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

Alternative 12: 47.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -9.2e-18)
   (/ a (* k k))
   (if (<= m 1.15e+39) (/ a (fma 10.0 k 1.0)) (* (* a k) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -9.2e-18) {
		tmp = a / (k * k);
	} else if (m <= 1.15e+39) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -9.2e-18)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.15e+39)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(a * k) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -9.2e-18], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15e+39], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -9.2000000000000004e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 55.7%

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

   if -9.2000000000000004e-18 < m < 1.15000000000000006e39

   1. Initial program 89.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 61.8%

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

   if 1.15000000000000006e39 < m

   1. Initial program 80.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 9.6%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 17.9%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.45:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.2e-20)
   (/ a (* k k))
   (if (<= m 0.45) (fma (* -10.0 a) k a) (* (* a k) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.2e-20) {
		tmp = a / (k * k);
	} else if (m <= 0.45) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.2e-20)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.45)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = Float64(Float64(a * k) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.2e-20], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.45], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.19999999999999996e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in k around -inf

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

   10. Applied rewrites 55.7%

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

   if -1.19999999999999996e-20 < m < 0.450000000000000011

   1. Initial program 92.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 59.8%

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

   13. Applied rewrites 59.8%

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

   if 0.450000000000000011 < m

   1. Initial program 78.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 9.0%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 16.8%

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

Alternative 14: 25.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-306}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5.4e-306) (* (* a k) -10.0) (fma (* 10.0 a) k a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5.4e-306) {
		tmp = (a * k) * -10.0;
	} else {
		tmp = fma((10.0 * a), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5.4e-306)
		tmp = Float64(Float64(a * k) * -10.0);
	else
		tmp = fma(Float64(10.0 * a), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5.4e-306], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision], N[(N[(10.0 * a), $MachinePrecision] * k + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.40000000000000018e-306

   1. Initial program 90.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 15.1

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

   5. Applied rewrites 15.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 10.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 8.7%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 13.4%

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

   if 5.40000000000000018e-306 < k

   1. Initial program 90.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 54.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 33.3%

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

Alternative 15: 8.2% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, k, m):
	return (a * k) * -10.0

Julia

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

MATLAB

function tmp = code(a, k, m)
	tmp = (a * k) * -10.0;
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 39.7

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

5. Applied rewrites 39.7%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 28.0%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 22.0%

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

12. Taylor expanded in k around inf

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

14. Applied rewrites 7.3%

    \[\leadsto \left(a \cdot k\right) \cdot -10 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))