Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 97.5%

Time: 5.1s

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: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.3:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10, k, 1\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.3)
   (* a (/ (pow k m) (+ (fma 10.0 k 1.0) (* k k))))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.3) {
		tmp = a * (pow(k, m) / (fma(10.0, k, 1.0) + (k * k)));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.3)
		tmp = Float64(a * Float64((k ^ m) / Float64(fma(10.0, k, 1.0) + Float64(k * k))));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.3], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 * k + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.2999999999999998

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 94.1

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

   6. Applied rewrites 94.1%

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

   if 3.2999999999999998 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.3:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.3) (* a (/ (pow k m) (fma (+ 10.0 k) k 1.0))) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.3) {
		tmp = a * (pow(k, m) / fma((10.0 + k), k, 1.0));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.3)
		tmp = Float64(a * Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.3], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.2999999999999998

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 3.2999999999999998 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-13} \lor \neg \left(m \leq 2.7 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.1e-13) (not (<= m 2.7e-7)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.1e-13) || !(m <= 2.7e-7)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3.1e-13) || !(m <= 2.7e-7))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3.1e-13], N[Not[LessEqual[m, 2.7e-7]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.0999999999999999e-13 or 2.70000000000000009e-7 < m

   1. Initial program 89.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -3.0999999999999999e-13 < m < 2.70000000000000009e-7

   1. Initial program 88.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.9

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

   5. Applied rewrites 88.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-13} \lor \neg \left(m \leq 2.7 \cdot 10^{-7}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5800000000:\\ \;\;\;\;\frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-k, -99 \cdot a, -10 \cdot a\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5800000000.0)
   (/ (* (/ a (* k k)) 99.0) (* k k))
   (if (<= m 2.3)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (fma (- k) (* -99.0 a) (* -10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5800000000.0) {
		tmp = ((a / (k * k)) * 99.0) / (k * k);
	} else if (m <= 2.3) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(-k, (-99.0 * a), (-10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5800000000.0)
		tmp = Float64(Float64(Float64(a / Float64(k * k)) * 99.0) / Float64(k * k));
	elseif (m <= 2.3)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(fma(Float64(-k), Float64(-99.0 * a), Float64(-10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5800000000.0], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-k) * N[(-99.0 * a), $MachinePrecision] + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.8e9

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in k around -inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k} + a}{{k}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k} \cdot -1 + a}{{k}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      7. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 + 1\right) \cdot \frac{a}{k} + \left(\mathsf{neg}\left(-10\right)\right) \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 + 1\right) \cdot \frac{a}{k} + 10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-100 + 1, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      13. pow2 N/A

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

   8. Applied rewrites 66.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{\color{blue}{k \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{99 \cdot \frac{a}{{k}^{2}}}{k \cdot k} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k} \]

       5. lift-*.f64 72.1

          \[\leadsto \frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k} \]

   11. Applied rewrites 72.1%

       \[\leadsto \frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k} \]

   if -5.8e9 < m < 2.2999999999999998

   1. Initial program 89.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 2.2999999999999998 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 29.7

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

   8. Applied rewrites 29.7%

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

Alternative 5: 61.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.3:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-k, -99 \cdot a, -10 \cdot a\right), k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+23)
   (* a (/ 1.0 (* k k)))
   (if (<= m 2.3)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (fma (- k) (* -99.0 a) (* -10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+23) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 2.3) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(fma(-k, (-99.0 * a), (-10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+23)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 2.3)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(fma(Float64(-k), Float64(-99.0 * a), Float64(-10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+23], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-k) * N[(-99.0 * a), $MachinePrecision] + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.0500000000000001e23

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 45.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{100} - {k}^{2}}{10 - k}, k, 1\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower--.f64 45.3

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

   9. Applied rewrites 45.3%

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

   10. Taylor expanded in k around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 59.7

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

   12. Applied rewrites 59.7%

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

   if -1.0500000000000001e23 < m < 2.2999999999999998

   1. Initial program 89.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 2.2999999999999998 < m

   1. Initial program 79.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 29.7

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

   8. Applied rewrites 29.7%

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

Alternative 6: 59.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+23)
   (* a (/ 1.0 (* k k)))
   (if (<= m 6.2e+18) (/ a (fma (+ 10.0 k) k 1.0)) (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+23) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 6.2e+18) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+23)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 6.2e+18)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+23], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+18], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.0500000000000001e23

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 45.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{100} - {k}^{2}}{10 - k}, k, 1\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower--.f64 45.3

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

   9. Applied rewrites 45.3%

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

   10. Taylor expanded in k around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 59.7

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

   12. Applied rewrites 59.7%

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

   if -1.0500000000000001e23 < m < 6.2e18

   1. Initial program 88.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 6.2e18 < m

   1. Initial program 79.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 24.8

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

   11. Applied rewrites 24.8%

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

Alternative 7: 59.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+23)
   (/ a (* k k))
   (if (<= m 6.2e+18) (/ a (fma (+ 10.0 k) k 1.0)) (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+23) {
		tmp = a / (k * k);
	} else if (m <= 6.2e+18) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+23)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 6.2e+18)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+23], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+18], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.0500000000000001e23

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      6. flip-+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      8. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      12. pow2 N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      13. pow2 N/A

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

      14. lift-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if -1.0500000000000001e23 < m < 6.2e18

   1. Initial program 88.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 6.2e18 < m

   1. Initial program 79.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 24.8

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

   11. Applied rewrites 24.8%

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

Alternative 8: 58.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+23)
   (/ a (* k k))
   (if (<= m 6.2e+18) (/ a (fma k k 1.0)) (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+23) {
		tmp = a / (k * k);
	} else if (m <= 6.2e+18) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+23)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 6.2e+18)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+23], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+18], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.0500000000000001e23

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      6. flip-+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      8. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      12. pow2 N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      13. pow2 N/A

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

      14. lift-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if -1.0500000000000001e23 < m < 6.2e18

   1. Initial program 88.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 82.9%

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

   if 6.2e18 < m

   1. Initial program 79.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 24.8

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

   11. Applied rewrites 24.8%

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

Alternative 9: 48.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5800000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5800000000.0)
   (/ a (* k k))
   (if (<= m 6.2e+18) (/ a (fma 10.0 k 1.0)) (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5800000000.0) {
		tmp = a / (k * k);
	} else if (m <= 6.2e+18) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5800000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 6.2e+18)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5800000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+18], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.8e9

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      6. flip-+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      8. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      12. pow2 N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      13. pow2 N/A

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

      14. lift-*.f64 57.5

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

   8. Applied rewrites 57.5%

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

   if -5.8e9 < m < 6.2e18

   1. Initial program 88.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 65.2%

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

   if 6.2e18 < m

   1. Initial program 79.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 24.8

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

   11. Applied rewrites 24.8%

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

Alternative 10: 47.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -7 \cdot 10^{-304} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -7e-304) (not (<= k 0.1))) (/ a (* k k)) (fma (* -10.0 a) k a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -7e-304) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = fma((-10.0 * a), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -7e-304) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = fma(Float64(-10.0 * a), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -7e-304], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -7e-304 or 0.10000000000000001 < k

   1. Initial program 83.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      6. flip-+ N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      8. +-commutative N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      11. +-commutative N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      12. pow2 N/A

          \[\leadsto \frac{a}{{k}^{2}} \]

      13. pow2 N/A

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

      14. lift-*.f64 44.1

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

   8. Applied rewrites 44.1%

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

   if -7e-304 < k < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 57.1

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

   10. Applied rewrites 57.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -7 \cdot 10^{-304} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 6.2e+18) a (* (* k a) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.2e+18) {
		tmp = a;
	} else {
		tmp = (k * a) * -10.0;
	}
	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 (m <= 6.2d+18) then
        tmp = a
    else
        tmp = (k * a) * (-10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.2e+18) {
		tmp = a;
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 6.2e+18:
		tmp = a
	else:
		tmp = (k * a) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6.2e+18)
		tmp = a;
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 6.2e+18)
		tmp = a;
	else
		tmp = (k * a) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6.2e+18], a, N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.2e18

   1. Initial program 93.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto a \]
   7. Step-by-step derivation

   8. Applied rewrites 31.4%

      \[\leadsto a \]

   if 6.2e18 < m

   1. Initial program 79.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 11.8

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

   8. Applied rewrites 11.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 24.8

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

   11. Applied rewrites 24.8%

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

Alternative 12: 20.2% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 47.8

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

5. Applied rewrites 47.8%

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

6. Taylor expanded in k around 0

   \[\leadsto a \]
7. Step-by-step derivation

8. Applied rewrites 22.9%

   \[\leadsto a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))