Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.9%

Time: 4.4s

Alternatives: 12

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% 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: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   if 1 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

Alternative 2: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{{k}^{m}}}{a}}\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{1}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{10}{k} - -1}{{k}^{\left(m - 2\right)} \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e-17)
   (/ 1.0 (/ (/ (fma (- k -10.0) k 1.0) (pow k m)) a))
   (if (<= m 1.4e-9)
     (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ 1.0 (fma 10.0 k 1.0)))
     (/ 1.0 (/ (- (/ 10.0 k) -1.0) (* (pow k (- m 2.0)) a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e-17) {
		tmp = 1.0 / ((fma((k - -10.0), k, 1.0) / pow(k, m)) / a);
	} else if (m <= 1.4e-9) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / fma(10.0, k, 1.0));
	} else {
		tmp = 1.0 / (((10.0 / k) - -1.0) / (pow(k, (m - 2.0)) * a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e-17)
		tmp = Float64(1.0 / Float64(Float64(fma(Float64(k - -10.0), k, 1.0) / (k ^ m)) / a));
	elseif (m <= 1.4e-9)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64(1.0 / fma(10.0, k, 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(10.0 / k) - -1.0) / Float64((k ^ Float64(m - 2.0)) * a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e-17], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.4e-9], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(10.0 / k), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3999999999999999e-17

   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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   if -1.3999999999999999e-17 < m < 1.39999999999999992e-9

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 1.39999999999999992e-9 < 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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.3

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in k around inf

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

      1. lower-/.f64 75.6

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

   10. Applied rewrites 75.6%

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

Alternative 3: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. sum-to-mult N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 4: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. sum-to-mult N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-fma.f64 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. div-flip N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. add-flip N/A

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

   17. metadata-eval N/A

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

   18. sub-to-fraction-rev N/A

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

   19. lower--.f64 N/A

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

   20. lower-/.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{{k}^{m}}}{a}}\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{1}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e-17)
   (/ 1.0 (/ (/ (fma (- k -10.0) k 1.0) (pow k m)) a))
   (if (<= m 0.65)
     (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ 1.0 (fma 10.0 k 1.0)))
     (/ 1.0 (/ (pow k (- m)) a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e-17) {
		tmp = 1.0 / ((fma((k - -10.0), k, 1.0) / pow(k, m)) / a);
	} else if (m <= 0.65) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / fma(10.0, k, 1.0));
	} else {
		tmp = 1.0 / (pow(k, -m) / a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e-17)
		tmp = Float64(1.0 / Float64(Float64(fma(Float64(k - -10.0), k, 1.0) / (k ^ m)) / a));
	elseif (m <= 0.65)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64(1.0 / fma(10.0, k, 1.0)));
	else
		tmp = Float64(1.0 / Float64((k ^ Float64(-m)) / a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e-17], N[(1.0 / N[(N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.65], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3999999999999999e-17

   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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   if -1.3999999999999999e-17 < m < 0.650000000000000022

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 0.650000000000000022 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{1}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e-17)
   (* (/ (pow k m) (fma (- k -10.0) k 1.0)) a)
   (if (<= m 0.65)
     (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ 1.0 (fma 10.0 k 1.0)))
     (/ 1.0 (/ (pow k (- m)) a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e-17) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * a;
	} else if (m <= 0.65) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / fma(10.0, k, 1.0));
	} else {
		tmp = 1.0 / (pow(k, -m) / a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e-17)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * a);
	elseif (m <= 0.65)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64(1.0 / fma(10.0, k, 1.0)));
	else
		tmp = Float64(1.0 / Float64((k ^ Float64(-m)) / a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e-17], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.65], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.3999999999999999e-17

   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. lower-fma.f64 N/A

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

      15. lower-+.f64 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. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.3

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 90.3

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

   5. Applied rewrites 90.3%

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

   if -1.3999999999999999e-17 < m < 0.650000000000000022

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 0.650000000000000022 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{if}\;m \leq -5.1 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{1}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ (pow k (- m)) a))))
   (if (<= m -5.1e-17)
     t_0
     (if (<= m 0.65)
       (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ 1.0 (fma 10.0 k 1.0)))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 / (pow(k, -m) / a);
	double tmp;
	if (m <= -5.1e-17) {
		tmp = t_0;
	} else if (m <= 0.65) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / fma(10.0, k, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(1.0 / Float64((k ^ Float64(-m)) / a))
	tmp = 0.0
	if (m <= -5.1e-17)
		tmp = t_0;
	elseif (m <= 0.65)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64(1.0 / fma(10.0, k, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.1e-17], t$95$0, If[LessEqual[m, 0.65], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.1000000000000003e-17 or 0.650000000000000022 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

   if -5.1000000000000003e-17 < m < 0.650000000000000022

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

Alternative 8: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{if}\;m \leq -5.1 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k, m, 1\right) \cdot a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ (pow k (- m)) a))))
   (if (<= m -5.1e-17)
     t_0
     (if (<= m 0.65)
       (/ (* (fma (log k) m 1.0) a) (fma (- k -10.0) k 1.0))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 / (pow(k, -m) / a);
	double tmp;
	if (m <= -5.1e-17) {
		tmp = t_0;
	} else if (m <= 0.65) {
		tmp = (fma(log(k), m, 1.0) * a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(1.0 / Float64((k ^ Float64(-m)) / a))
	tmp = 0.0
	if (m <= -5.1e-17)
		tmp = t_0;
	elseif (m <= 0.65)
		tmp = Float64(Float64(fma(log(k), m, 1.0) * a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.1e-17], t$95$0, If[LessEqual[m, 0.65], N[(N[(N[(N[Log[k], $MachinePrecision] * m + 1.0), $MachinePrecision] * a), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.1000000000000003e-17 or 0.650000000000000022 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

   if -5.1000000000000003e-17 < m < 0.650000000000000022

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. sum-to-mult-rev N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. mult-flip N/A

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

      17. lift-/.f64 90.3

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 90.3

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

      21. lift-fma.f64 N/A

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

      22. lift-fma.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 40.7

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

   8. Applied rewrites 40.7%

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

Alternative 9: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{if}\;m \leq -5.1 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ (pow k (- m)) a))))
   (if (<= m -5.1e-17)
     t_0
     (if (<= m 0.65) (/ a (fma (- k -10.0) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 / (pow(k, -m) / a);
	double tmp;
	if (m <= -5.1e-17) {
		tmp = t_0;
	} else if (m <= 0.65) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(1.0 / Float64((k ^ Float64(-m)) / a))
	tmp = 0.0
	if (m <= -5.1e-17)
		tmp = t_0;
	elseif (m <= 0.65)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.1e-17], t$95$0, If[LessEqual[m, 0.65], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.1000000000000003e-17 or 0.650000000000000022 < 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. 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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

   if -5.1000000000000003e-17 < m < 0.650000000000000022

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. sum-to-mult-rev N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. mult-flip N/A

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

      17. lift-/.f64 90.3

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 90.3

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

      21. lift-fma.f64 N/A

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

      22. lift-fma.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift--.f64 45.3

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

   8. Applied rewrites 45.3%

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

Alternative 10: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0125:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\left(-\log k\right) \cdot \left(m - 2\right)} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.0125)
   (/ 1.0 (/ (pow k (- m)) a))
   (* (exp (- (* (- (log k)) (- m 2.0)))) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.0125) {
		tmp = 1.0 / (pow(k, -m) / a);
	} else {
		tmp = exp(-(-log(k) * (m - 2.0))) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.0125d0) then
        tmp = 1.0d0 / ((k ** -m) / a)
    else
        tmp = exp(-(-log(k) * (m - 2.0d0))) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.0125) {
		tmp = 1.0 / (Math.pow(k, -m) / a);
	} else {
		tmp = Math.exp(-(-Math.log(k) * (m - 2.0))) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.0125:
		tmp = 1.0 / (math.pow(k, -m) / a)
	else:
		tmp = math.exp(-(-math.log(k) * (m - 2.0))) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.0125)
		tmp = Float64(1.0 / Float64((k ^ Float64(-m)) / a));
	else
		tmp = Float64(exp(Float64(-Float64(Float64(-log(k)) * Float64(m - 2.0)))) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.0125)
		tmp = 1.0 / ((k ^ -m) / a);
	else
		tmp = exp(-(-log(k) * (m - 2.0))) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.0125], N[(1.0 / N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-N[((-N[Log[k], $MachinePrecision]) * N[(m - 2.0), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.012500000000000001

   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-/.f64 N/A

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

      4. lower-/.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. lower-fma.f64 N/A

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

      13. lower-+.f64 90.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. 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-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 90.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

      1. pow-flip N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-neg.f64 83.2

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

   8. Applied rewrites 83.2%

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

   if 0.012500000000000001 < 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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 80.4

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{k}\right) \cdot \left(m - 2\right)\right)} \cdot a \]

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

         \[\leadsto e^{-\left(\mathsf{neg}\left(\log k\right)\right) \cdot \left(m - 2\right)} \cdot a \]

      8. lower-neg.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lift--.f64 50.8

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

   10. Applied rewrites 50.8%

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

Alternative 11: 46.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{a}{k \cdot k}}{\frac{10 - \frac{-1}{k}}{k} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.6e+30)
   (/ (/ a (* k k)) (- (/ (- 10.0 (/ -1.0 k)) k) -1.0))
   (/ a (fma (- k -10.0) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.6e+30) {
		tmp = (a / (k * k)) / (((10.0 - (-1.0 / k)) / k) - -1.0);
	} else {
		tmp = a / fma((k - -10.0), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.6e+30)
		tmp = Float64(Float64(a / Float64(k * k)) / Float64(Float64(Float64(10.0 - Float64(-1.0 / k)) / k) - -1.0));
	else
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.6e+30], N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.59999999999999986e30

   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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 80.4

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in m around 0

      \[\leadsto \frac{\color{blue}{\frac{a}{{k}^{2}}}}{\frac{10 - \frac{-1}{k}}{k} - -1} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{a}{\color{blue}{{k}^{2}}}}{\frac{10 - \frac{-1}{k}}{k} - -1} \]

      2. unpow2 N/A

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

      3. lower-*.f64 35.3

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

   10. Applied rewrites 35.3%

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

   if -1.59999999999999986e30 < 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

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. sum-to-mult-rev N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. mult-flip N/A

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

      17. lift-/.f64 90.3

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 90.3

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

      21. lift-fma.f64 N/A

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

      22. lift-fma.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift--.f64 45.3

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

   8. Applied rewrites 45.3%

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

Alternative 12: 45.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (/ a (fma (- k -10.0) k 1.0)))

C

double code(double a, double k, double m) {
	return a / fma((k - -10.0), k, 1.0);
}

Julia

function code(a, k, m)
	return Float64(a / fma(Float64(k - -10.0), k, 1.0))
end

Wolfram

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

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

   3. sum-to-mult N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. sum-to-mult-rev N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 N/A

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

   12. associate-*r/ N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. mult-flip N/A

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

   17. lift-/.f64 90.3

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 90.3

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

   21. lift-fma.f64 N/A

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

   22. lift-fma.f64 N/A

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

5. Applied rewrites 90.3%

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

6. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-flip N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f64 N/A

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

   8. lift--.f64 45.3

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

8. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))