Falkner and Boettcher, Appendix A

Percentage Accurate: 91.0% → 97.5%

Time: 4.8s

Alternatives: 10

Speedup: 1.3×

• 22.1% 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: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+208}:\\ \;\;\;\;{k}^{m} \cdot \frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

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))) 5e+208)
    (* (pow k m) (/ a_m (fma (+ 10.0 k) k 1.0)))
    (* (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))) <= 5e+208) {
		tmp = pow(k, m) * (a_m / fma((10.0 + k), k, 1.0));
	} 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))) <= 5e+208)
		tmp = Float64((k ^ m) * Float64(a_m / fma(Float64(10.0 + k), k, 1.0)));
	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], 5e+208], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $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))) < 5.0000000000000004e208

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 91.0

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. pow2 N/A

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

      11. associate-+l+ N/A

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

      12. pow2 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 91.0

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

   3. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 5.0000000000000004e208 < (/.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 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around 0

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

      1. sqr-pow N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-pow N/A

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

      7. lower-pow.f64 82.7

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

   10. Applied rewrites 82.7%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-10}:\\ \;\;\;\;{k}^{m} \cdot \frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a\_m \cdot m, \log k, a\_m\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

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 (<= m -5.4e-10)
    (* (pow k m) (/ a_m (* k k)))
    (if (<= m 5e-7)
      (/ (fma (* a_m m) (log k) a_m) (fma (+ 10.0 k) k 1.0))
      (* (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 (m <= -5.4e-10) {
		tmp = pow(k, m) * (a_m / (k * k));
	} else if (m <= 5e-7) {
		tmp = fma((a_m * m), log(k), a_m) / fma((10.0 + k), k, 1.0);
	} 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 (m <= -5.4e-10)
		tmp = Float64((k ^ m) * Float64(a_m / Float64(k * k)));
	elseif (m <= 5e-7)
		tmp = Float64(fma(Float64(a_m * m), log(k), a_m) / fma(Float64(10.0 + k), k, 1.0));
	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[m, -5.4e-10], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-7], N[(N[(N[(a$95$m * m), $MachinePrecision] * N[Log[k], $MachinePrecision] + a$95$m), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -5.4e-10

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 91.0

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. pow2 N/A

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

      11. associate-+l+ N/A

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

      12. pow2 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 91.0

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

   3. Applied rewrites 91.0%

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

   4. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 65.4

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

   6. Applied rewrites 65.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -5.4e-10 < m < 4.99999999999999977e-7

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 40.5

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

   6. Applied rewrites 40.5%

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

   if 4.99999999999999977e-7 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around 0

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

      1. sqr-pow N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-pow N/A

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

      7. lower-pow.f64 82.7

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

   10. Applied rewrites 82.7%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a\_m \cdot m, \log k, a\_m\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -8.5e-5)
      t_0
      (if (<= m 5e-7)
        (/ (fma (* a_m m) (log k) a_m) (fma (+ 10.0 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 <= -8.5e-5) {
		tmp = t_0;
	} else if (m <= 5e-7) {
		tmp = fma((a_m * m), log(k), a_m) / fma((10.0 + 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 <= -8.5e-5)
		tmp = t_0;
	elseif (m <= 5e-7)
		tmp = Float64(fma(Float64(a_m * m), log(k), a_m) / fma(Float64(10.0 + 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, -8.5e-5], t$95$0, If[LessEqual[m, 5e-7], N[(N[(N[(a$95$m * m), $MachinePrecision] * N[Log[k], $MachinePrecision] + a$95$m), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.500000000000001e-5 or 4.99999999999999977e-7 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around 0

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

      1. sqr-pow N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-pow N/A

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

      7. lower-pow.f64 82.7

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

   10. Applied rewrites 82.7%

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

   if -8.500000000000001e-5 < m < 4.99999999999999977e-7

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 40.5

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

   6. Applied rewrites 40.5%

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

Alternative 4: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\_m\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -8.5e-5)
      t_0
      (if (<= m 4e-9) (/ a_m (fma (+ 10.0 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 <= -8.5e-5) {
		tmp = t_0;
	} else if (m <= 4e-9) {
		tmp = a_m / fma((10.0 + 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 <= -8.5e-5)
		tmp = t_0;
	elseif (m <= 4e-9)
		tmp = Float64(a_m / fma(Float64(10.0 + 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, -8.5e-5], t$95$0, If[LessEqual[m, 4e-9], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.500000000000001e-5 or 4.00000000000000025e-9 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around 0

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

      1. sqr-pow N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-pow N/A

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

      7. lower-pow.f64 82.7

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

   10. Applied rewrites 82.7%

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

   if -8.500000000000001e-5 < m < 4.00000000000000025e-9

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

Alternative 5: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.22:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

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 -0.22)
    (/ a_m (* k k))
    (if (<= m 1.3) (/ a_m (fma (+ 10.0 k) k 1.0)) (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.22) {
		tmp = a_m / (k * k);
	} else if (m <= 1.3) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	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 <= -0.22)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 1.3)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	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_] := N[(a$95$s * If[LessEqual[m, -0.22], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.220000000000000001

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   if -0.220000000000000001 < m < 1.30000000000000004

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if 1.30000000000000004 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   8. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

      3. *-commutative N/A

         \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

      5. pow2 N/A

         \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

      6. lift-*.f64 22.3

         \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   10. Applied rewrites 22.3%

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

Alternative 6: 61.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -5.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

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 -5.9e-18)
    (/ a_m (* k k))
    (if (<= m 1.3) (/ a_m (fma 10.0 k 1.0)) (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -5.9e-18) {
		tmp = a_m / (k * k);
	} else if (m <= 1.3) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	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 <= -5.9e-18)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 1.3)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	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_] := N[(a$95$s * If[LessEqual[m, -5.9e-18], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -5.90000000000000019e-18

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   if -5.90000000000000019e-18 < m < 1.30000000000000004

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 28.3%

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

   if 1.30000000000000004 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   8. Taylor expanded in k around inf

      \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

      3. *-commutative N/A

         \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

      5. pow2 N/A

         \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

      6. lift-*.f64 22.3

         \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   10. Applied rewrites 22.3%

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

Alternative 7: 47.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ t_1 := \frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;\frac{a\_m}{1}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* k k)))
        (t_1 (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    a_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 1e+299)
        (/ a_m 1.0)
        (if (<= t_1 INFINITY) t_0 (fma (* -10.0 a_m) k 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 / (k * k);
	double t_1 = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+299) {
		tmp = a_m / 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma((-10.0 * a_m), k, 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(a_m / Float64(k * k))
	t_1 = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+299)
		tmp = Float64(a_m / 1.0);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(-10.0 * a_m), k, 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[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+299], N[(a$95$m / 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(-10.0 * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 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 or 1.0000000000000001e299 < (/.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 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e299

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

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

   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 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 21.0

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

   10. Applied rewrites 21.0%

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

Alternative 8: 46.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -5.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a\_m, k, a\_m\right)\\ \end{array} \end{array} \]

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 -5.9e-18)
    (/ a_m (* k k))
    (if (<= m 1.3e+31) (/ a_m (fma 10.0 k 1.0)) (fma (* -10.0 a_m) k 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 (m <= -5.9e-18) {
		tmp = a_m / (k * k);
	} else if (m <= 1.3e+31) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else {
		tmp = fma((-10.0 * a_m), k, 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 (m <= -5.9e-18)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 1.3e+31)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	else
		tmp = fma(Float64(-10.0 * a_m), k, 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[m, -5.9e-18], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3e+31], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -5.90000000000000019e-18

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   if -5.90000000000000019e-18 < m < 1.3e31

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 28.3%

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

   if 1.3e31 < m

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 21.0

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

   10. Applied rewrites 21.0%

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

Alternative 9: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ t_1 := \frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;\frac{a\_m}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (/ a_m (* k k)))
        (t_1 (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (* a_s (if (<= t_1 0.0) t_0 (if (<= t_1 1e+299) (/ a_m 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 = a_m / (k * k);
	double t_1 = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+299) {
		tmp = a_m / 1.0;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a_m / (k * k)
    t_1 = (a_m * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+299) then
        tmp = a_m / 1.0d0
    else
        tmp = t_0
    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 t_0 = a_m / (k * k);
	double t_1 = (a_m * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+299) {
		tmp = a_m / 1.0;
	} else {
		tmp = t_0;
	}
	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):
	t_0 = a_m / (k * k)
	t_1 = (a_m * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 1e+299:
		tmp = a_m / 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(a_m / Float64(k * k))
	t_1 = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+299)
		tmp = Float64(a_m / 1.0);
	else
		tmp = t_0;
	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)
	t_0 = a_m / (k * k);
	t_1 = (a_m * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+299)
		tmp = a_m / 1.0;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+299], N[(a$95$m / 1.0), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0 or 1.0000000000000001e299 < (/.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 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e299

   1. Initial program 91.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 20.2%

      \[\leadsto \frac{a}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 20.2% 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 91.0%

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

2. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 45.0

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

4. Applied rewrites 45.0%

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

5. Taylor expanded in k around 0

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

7. Applied rewrites 20.2%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))