Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 97.7%

Time: 10.8s

Alternatives: 15

Speedup: 1.1×

• 22.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.21)
   (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.21d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.21:
		tmp = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.21)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.21)
		tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.21], 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], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.209999999999999992

   1. Initial program 95.8%

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

   if 0.209999999999999992 < m

   1. Initial program 73.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 2: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{{k}^{-1} - -10}{k}, k, k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{{k}^{-1}}{k} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* (fma (/ (- (pow k -1.0) -10.0) k) k k) k))
     (if (<= t_0 5e+281)
       (/ a (fma (- k -10.0) k 1.0))
       (if (<= t_0 INFINITY)
         (* (/ (pow k -1.0) k) a)
         (* (fma (fma 99.0 k -10.0) k 1.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (fma(((pow(k, -1.0) - -10.0) / k), k, k) * k);
	} else if (t_0 <= 5e+281) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (pow(k, -1.0) / k) * a;
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(fma(Float64(Float64((k ^ -1.0) - -10.0) / k), k, k) * k));
	elseif (t_0 <= 5e+281)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64((k ^ -1.0) / k) * a);
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] - -10.0), $MachinePrecision] / k), $MachinePrecision] * k + k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+281], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Power[k, -1.0], $MachinePrecision] / k), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 95.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   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 31.1%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.3%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e281

   1. Initial program 99.9%

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if 5.00000000000000016e281 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 3.5

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 52.8%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 0.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{{k}^{-1} - -10}{k}, k, k\right) \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{{k}^{-1}}{k} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{{k}^{-1} - -10}{k}, k, k\right) \cdot k}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{99}{k} + -10}{k} + 1}{k \cdot k} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (* (fma (/ (- (pow k -1.0) -10.0) k) k k) k))
     (if (<= t_0 5e+281)
       (/ a (fma (- k -10.0) k 1.0))
       (if (<= t_0 INFINITY)
         (* (/ (+ (/ (+ (/ 99.0 k) -10.0) k) 1.0) (* k k)) a)
         (* (fma (fma 99.0 k -10.0) k 1.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / (fma(((pow(k, -1.0) - -10.0) / k), k, k) * k);
	} else if (t_0 <= 5e+281) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((((99.0 / k) + -10.0) / k) + 1.0) / (k * k)) * a;
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / Float64(fma(Float64(Float64((k ^ -1.0) - -10.0) / k), k, k) * k));
	elseif (t_0 <= 5e+281)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(99.0 / k) + -10.0) / k) + 1.0) / Float64(k * k)) * a);
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] - -10.0), $MachinePrecision] / k), $MachinePrecision] * k + k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+281], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[(N[(99.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 95.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   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 31.1%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.3%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e281

   1. Initial program 99.9%

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if 5.00000000000000016e281 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 3.5

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{\left(1 + \frac{99}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \frac{\frac{\frac{99}{k} + -10}{k} + 1}{\color{blue}{k \cdot k}} \cdot a \]

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 0.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(\frac{{k}^{-1} - -10}{k}, k, k\right) \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{99}{k} + -10}{k} + 1}{k \cdot k} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_1 4e-279)
     t_0
     (if (<= t_1 5e+281)
       (/ a (fma 10.0 k 1.0))
       (if (<= t_1 INFINITY) t_0 (* (fma (fma 99.0 k -10.0) k 1.0) a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 4e-279) {
		tmp = t_0;
	} else if (t_1 <= 5e+281) {
		tmp = a / fma(10.0, k, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 4e-279)
		tmp = t_0;
	elseif (t_1 <= 5e+281)
		tmp = Float64(a / fma(10.0, k, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-279], t$95$0, If[LessEqual[t$95$1, 5e+281], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000022e-279 or 5.00000000000000016e281 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 96.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 48.2%

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

   if 4.00000000000000022e-279 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e281

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   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 65.6%

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

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

   1. Initial program 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 0.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 100.0%

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

Alternative 5: 40.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-279} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (or (<= t_0 4e-279) (not (<= t_0 5e+281)))
     (/ a (* k k))
     (/ a (fma 10.0 k 1.0)))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if ((t_0 <= 4e-279) || !(t_0 <= 5e+281)) {
		tmp = a / (k * k);
	} else {
		tmp = a / fma(10.0, k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if ((t_0 <= 4e-279) || !(t_0 <= 5e+281))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / fma(10.0, k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-279], N[Not[LessEqual[t$95$0, 5e+281]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000022e-279 or 5.00000000000000016e281 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 88.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 44.2%

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

   if 4.00000000000000022e-279 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e281

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   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 65.6%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{-279} \lor \neg \left(\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-279} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (or (<= t_0 4e-279) (not (<= t_0 5e+281))) (/ a (* k k)) (/ a 1.0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if ((t_0 <= 4e-279) || !(t_0 <= 5e+281)) {
		tmp = a / (k * k);
	} else {
		tmp = a / 1.0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    if ((t_0 <= 4d-279) .or. (.not. (t_0 <= 5d+281))) then
        tmp = a / (k * k)
    else
        tmp = a / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if ((t_0 <= 4e-279) || !(t_0 <= 5e+281)) {
		tmp = a / (k * k);
	} else {
		tmp = a / 1.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if (t_0 <= 4e-279) or not (t_0 <= 5e+281):
		tmp = a / (k * k)
	else:
		tmp = a / 1.0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if ((t_0 <= 4e-279) || !(t_0 <= 5e+281))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if ((t_0 <= 4e-279) || ~((t_0 <= 5e+281)))
		tmp = a / (k * k);
	else
		tmp = a / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-279], N[Not[LessEqual[t$95$0, 5e+281]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000022e-279 or 5.00000000000000016e281 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 88.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 44.2%

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

   if 4.00000000000000022e-279 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e281

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 65.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{-279} \lor \neg \left(\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+281}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.21) (* (/ (pow k m) (fma k (+ 10.0 k) 1.0)) a) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = (pow(k, m) / fma(k, (10.0 + k), 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.21)
		tmp = Float64(Float64((k ^ m) / fma(k, Float64(10.0 + k), 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.21], N[(N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(10.0 + k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.209999999999999992

   1. Initial program 95.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.8

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if 0.209999999999999992 < m

   1. Initial program 73.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 8: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{{k}^{-1}}{k} \cdot a\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.29)
   (* (/ (pow k -1.0) k) a)
   (if (<= m 0.21)
     (/ a (fma (- k -10.0) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.29) {
		tmp = (pow(k, -1.0) / k) * a;
	} else if (m <= 0.21) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.29)
		tmp = Float64(Float64((k ^ -1.0) / k) * a);
	elseif (m <= 0.21)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.29], N[(N[(N[Power[k, -1.0], $MachinePrecision] / k), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.21], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.28999999999999998

   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 43.0

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

   7. Applied rewrites 43.0%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 77.2%

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

   if -0.28999999999999998 < m < 0.209999999999999992

   1. Initial program 92.3%

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 0.209999999999999992 < m

   1. Initial program 73.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 30.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), \color{blue}{k}, 1\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{{k}^{-1}}{k} \cdot a\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.4 \cdot 10^{-7} \lor \neg \left(m \leq 0.00045\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -7.4e-7) (not (<= m 0.00045)))
   (* (pow k m) a)
   (/ a (fma (- k -10.0) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -7.4e-7) || !(m <= 0.00045)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((k - -10.0), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -7.4e-7) || !(m <= 0.00045))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -7.4e-7], N[Not[LessEqual[m, 0.00045]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -7.40000000000000009e-7 or 4.4999999999999999e-4 < m

   1. Initial program 87.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.5

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -7.40000000000000009e-7 < m < 4.4999999999999999e-4

   1. Initial program 92.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.4 \cdot 10^{-7} \lor \neg \left(m \leq 0.00045\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.29:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.21:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.29)
   (/ a (* k k))
   (if (<= m 0.21)
     (/ a (fma (- k -10.0) k 1.0))
     (* (fma (fma 99.0 k -10.0) k 1.0) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.29) {
		tmp = a / (k * k);
	} else if (m <= 0.21) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.29)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.21)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.29], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.21], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 76.1%

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

   if -0.28999999999999998 < m < 0.209999999999999992

   1. Initial program 92.3%

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

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 0.209999999999999992 < m

   1. Initial program 73.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 30.5%

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

Alternative 11: 24.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.2e+29) (/ a 1.0) (* (* k a) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.2e+29) {
		tmp = a / 1.0;
	} 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 <= 1.2d+29) then
        tmp = a / 1.0d0
    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 <= 1.2e+29) {
		tmp = a / 1.0;
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.2e+29:
		tmp = a / 1.0
	else:
		tmp = (k * a) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.2e+29)
		tmp = Float64(a / 1.0);
	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 <= 1.2e+29)
		tmp = a / 1.0;
	else
		tmp = (k * a) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.2e+29], N[(a / 1.0), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.2e29

   1. Initial program 94.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 26.0%

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

   if 1.2e29 < m

   1. Initial program 74.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 9.8%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 15.0%

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

   13. Applied rewrites 15.0%

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

Alternative 12: 20.4% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 21.2%

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

Alternative 13: 20.4% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 21.2%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 21.2%

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

Alternative 14: 8.6% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 21.2%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 5.4%

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

13. Applied rewrites 5.4%

    \[\leadsto \color{blue}{\left(k \cdot a\right) \cdot -10} \]
14. Add Preprocessing

Alternative 15: 8.6% accurate, 12.2× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	return (-10.0 * a) * 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 = ((-10.0d0) * a) * k
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 21.2%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 5.4%

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

Reproduce

herbie shell --seed 2024357 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))