Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.7%

Time: 6.1s

Alternatives: 12

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{k - -10}{a\_m}, k, \frac{1}{a\_m}\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 4e+286)
    (/ 1.0 (/ (fma (/ (- k -10.0) a_m) k (/ 1.0 a_m)) (pow k m)))
    (* (pow k m) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (((a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 4e+286) {
		tmp = 1.0 / (fma(((k - -10.0) / a_m), k, (1.0 / a_m)) / pow(k, m));
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 4e+286)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(k - -10.0) / a_m), k, Float64(1.0 / a_m)) / (k ^ m)));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+286], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] / a$95$m), $MachinePrecision] * k + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000013e286

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-special-/.f64 90.2

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 90.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.2

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

   3. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if 4.00000000000000013e286 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.5

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

   6. Applied rewrites 82.5%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+105}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{k}{a\_m}, k, \frac{1}{a\_m}\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e+105)
    (/ 1.0 (/ (fma (/ k a_m) k (/ 1.0 a_m)) (pow k m)))
    (* (pow k m) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (((a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e+105) {
		tmp = 1.0 / (fma((k / a_m), k, (1.0 / a_m)) / pow(k, m));
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e+105)
		tmp = Float64(1.0 / Float64(fma(Float64(k / a_m), k, Float64(1.0 / a_m)) / (k ^ m)));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+105], N[(1.0 / N[(N[(N[(k / a$95$m), $MachinePrecision] * k + N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.9999999999999994e104

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-special-/.f64 90.2

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 90.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.2

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

   3. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 91.0

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

   8. Applied rewrites 91.0%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.5

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

   6. Applied rewrites 82.5%

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

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 4e+286)
    (* (/ (pow k m) (fma (- k -10.0) k 1.0)) a_m)
    (* (pow k m) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (((a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 4e+286) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * a_m;
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 4e+286)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * a_m);
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+286], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000013e286

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   3. Applied rewrites 90.3%

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

   if 4.00000000000000013e286 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.5

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

   6. Applied rewrites 82.5%

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

Alternative 4: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.58:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m)))
   (*
    a_s
    (if (<= m -1.2e-11)
      t_0
      (if (<= m 0.58) (/ a_m (fma k 10.0 (fma k k 1.0))) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double tmp;
	if (m <= -1.2e-11) {
		tmp = t_0;
	} else if (m <= 0.58) {
		tmp = a_m / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	tmp = 0.0
	if (m <= -1.2e-11)
		tmp = t_0;
	elseif (m <= 0.58)
		tmp = Float64(a_m / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -1.2e-11], t$95$0, If[LessEqual[m, 0.58], N[(a$95$m / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.2000000000000001e-11 or 0.57999999999999996 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.5

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

   6. Applied rewrites 82.5%

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

   if -1.2000000000000001e-11 < m < 0.57999999999999996

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. *-commutative N/A

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

      15. add-flip-rev N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 44.6

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

   8. Applied rewrites 44.6%

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

Alternative 5: 59.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a\_m \cdot \frac{a\_m}{\mathsf{fma}\left(k - -10, k, 1\right)}}{a\_m}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, a\_m, \frac{a\_m}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{k \cdot \left(10 + -99 \cdot k\right) - 1}{\frac{-1}{a\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    a_s
    (if (<= t_0 0.0)
      (/ (* a_m (/ a_m (fma (- k -10.0) k 1.0))) a_m)
      (if (<= t_0 4e+291)
        (/ a_m (fma k 10.0 (fma k k 1.0)))
        (if (<= t_0 INFINITY)
          (* k (fma -10.0 a_m (/ a_m k)))
          (/ (- (* k (+ 10.0 (* -99.0 k))) 1.0) (/ -1.0 a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (a_m * (a_m / fma((k - -10.0), k, 1.0))) / a_m;
	} else if (t_0 <= 4e+291) {
		tmp = a_m / fma(k, 10.0, fma(k, k, 1.0));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = k * fma(-10.0, a_m, (a_m / k));
	} else {
		tmp = ((k * (10.0 + (-99.0 * k))) - 1.0) / (-1.0 / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(a_m * Float64(a_m / fma(Float64(k - -10.0), k, 1.0))) / a_m);
	elseif (t_0 <= 4e+291)
		tmp = Float64(a_m / fma(k, 10.0, fma(k, k, 1.0)));
	elseif (t_0 <= Inf)
		tmp = Float64(k * fma(-10.0, a_m, Float64(a_m / k)));
	else
		tmp = Float64(Float64(Float64(k * Float64(10.0 + Float64(-99.0 * k))) - 1.0) / Float64(-1.0 / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(a$95$m * N[(a$95$m / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[t$95$0, 4e+291], N[(a$95$m / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(k * N[(-10.0 * a$95$m + N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * N[(10.0 + N[(-99.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(-1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 43.5%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.9999999999999998e291

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. *-commutative N/A

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

      15. add-flip-rev N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if 3.9999999999999998e291 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

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

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in k around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 28.4

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

   11. Applied rewrites 28.4%

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

Alternative 6: 54.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.46:\\ \;\;\;\;\frac{a\_m \cdot \frac{a\_m}{\mathsf{fma}\left(k - -10, k, 1\right)}}{a\_m}\\ \mathbf{elif}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.46)
    (/ (* a_m (/ a_m (fma (- k -10.0) k 1.0))) a_m)
    (if (<= m 2.9) (/ a_m (fma k 10.0 (fma k k 1.0))) (* -10.0 (* a_m k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.46) {
		tmp = (a_m * (a_m / fma((k - -10.0), k, 1.0))) / a_m;
	} else if (m <= 2.9) {
		tmp = a_m / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.46)
		tmp = Float64(Float64(a_m * Float64(a_m / fma(Float64(k - -10.0), k, 1.0))) / a_m);
	elseif (m <= 2.9)
		tmp = Float64(a_m / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.46], N[(N[(a$95$m * N[(a$95$m / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[m, 2.9], N[(a$95$m / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.46000000000000002

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 43.5%

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

   if -0.46000000000000002 < m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. *-commutative N/A

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

      15. add-flip-rev N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 7: 52.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.74:\\ \;\;\;\;\frac{a\_m \cdot a\_m}{\mathsf{fma}\left(k - -10, k, 1\right) \cdot a\_m}\\ \mathbf{elif}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.74)
    (/ (* a_m a_m) (* (fma (- k -10.0) k 1.0) a_m))
    (if (<= m 2.9) (/ a_m (fma k 10.0 (fma k k 1.0))) (* -10.0 (* a_m k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.74) {
		tmp = (a_m * a_m) / (fma((k - -10.0), k, 1.0) * a_m);
	} else if (m <= 2.9) {
		tmp = a_m / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.74)
		tmp = Float64(Float64(a_m * a_m) / Float64(fma(Float64(k - -10.0), k, 1.0) * a_m));
	elseif (m <= 2.9)
		tmp = Float64(a_m / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.74], N[(N[(a$95$m * a$95$m), $MachinePrecision] / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.9], N[(a$95$m / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.73999999999999999

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   8. Applied rewrites 39.1%

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

   if -0.73999999999999999 < m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. *-commutative N/A

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

      15. add-flip-rev N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 8: 50.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.9) (/ a_m (fma k 10.0 (fma k k 1.0))) (* -10.0 (* a_m k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a_m / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a_m / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.9], N[(a$95$m / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. *-commutative N/A

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

      15. add-flip-rev N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 9: 50.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 2.9) (/ a_m (fma (- k -10.0) k 1.0)) (* -10.0 (* a_m k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a_m / fma((k - -10.0), k, 1.0);
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a_m / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.9], N[(a$95$m / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 10: 33.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 2.9) (/ a_m (fma 10.0 k 1.0)) (* -10.0 (* a_m k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.9], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 27.9%

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

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 11: 25.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 2.9) (/ a_m 1.0) (* -10.0 (* a_m k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a_m / 1.0;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.9d0) then
        tmp = a_m / 1.0d0
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a_m / 1.0;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 2.9:
		tmp = a_m / 1.0
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a_m / 1.0);
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 2.9)
		tmp = a_m / 1.0;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.9], N[(a$95$m / 1.0), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 19.6%

      \[\leadsto \frac{a}{1} \]

   if 2.89999999999999991 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 19.7

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

   10. Applied rewrites 19.7%

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

   11. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 8.2

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

   13. Applied rewrites 8.2%

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

Alternative 12: 19.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \frac{a\_m}{1} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (/ a_m 1.0)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / 1.0);
}

Fortran

a\_m =     private
a\_s =     private
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_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * (a_m / 1.0d0)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / 1.0);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (a_m / 1.0)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m / 1.0))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (a_m / 1.0);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(a$95$m / 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.3%

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

2. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 44.6

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

4. Applied rewrites 44.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

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

   7. sub-flip N/A

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

   8. lift--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 44.6

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

6. Applied rewrites 44.6%

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

7. Taylor expanded in k around 0

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

9. Applied rewrites 19.6%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))