Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 97.8%

Time: 8.9s

Alternatives: 12

Speedup: 1.0×

• 22.5% 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: 90.4% 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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.5) (* (/ (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 <= 3.5) {
		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 <= 3.5)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.5], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if 3.5 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 78.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 78.6

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

   4. Applied rewrites 78.6%

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

   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: 17.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
   (* (* -10.0 a) k)
   (fma (* -10.0 a) k a)))

C

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 0.0], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 11.7%

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

   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 11.6%

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

   13. Applied rewrites 11.6%

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

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

   1. Initial program 71.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 27.2%

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

   10. Applied rewrites 27.2%

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

Alternative 3: 16.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{-277}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 1e-277)
   (* (* -10.0 a) k)
   (* (fma -10.0 k 1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (((a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 1e-277) {
		tmp = (-10.0 * a) * k;
	} else {
		tmp = fma(-10.0, k, 1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 1e-277)
		tmp = Float64(Float64(-10.0 * a) * k);
	else
		tmp = Float64(fma(-10.0, k, 1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 1e-277], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 9.99999999999999969e-278

   1. Initial program 96.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 11.6%

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

   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 11.5%

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

   13. Applied rewrites 11.5%

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

   if 9.99999999999999969e-278 < (/.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 70.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 35.6

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 27.8%

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

   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 27.8%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.08 \cdot 10^{-13} \lor \neg \left(m \leq 0.0064\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.08e-13) (not (<= m 0.0064)))
   (* (pow k m) a)
   (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.08e-13) || !(m <= 0.0064)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.08e-13) || !(m <= 0.0064))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.08e-13], N[Not[LessEqual[m, 0.0064]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.0799999999999999e-13 or 0.00640000000000000031 < m

   1. Initial program 88.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 88.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -1.0799999999999999e-13 < m < 0.00640000000000000031

   1. Initial program 91.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   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. lower-+.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.08 \cdot 10^{-13} \lor \neg \left(m \leq 0.0064\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.1:\\ \;\;\;\;\frac{\frac{\frac{a}{k \cdot k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.1)
   (/ (/ (* (/ a (* k k)) 99.0) k) k)
   (if (<= m 1.1)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.1) {
		tmp = (((a / (k * k)) * 99.0) / k) / k;
	} else if (m <= 1.1) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.1)
		tmp = Float64(Float64(Float64(Float64(a / Float64(k * k)) * 99.0) / k) / k);
	elseif (m <= 1.1)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.1], N[(N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 1.1], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      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. lower-+.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 83.3%

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

   if -0.10000000000000001 < m < 1.1000000000000001

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.2

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

   4. Applied rewrites 92.2%

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

   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. lower-+.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 1.1000000000000001 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.1:\\ \;\;\;\;\frac{\frac{\frac{a}{k \cdot k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.165:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.165)
   (/ a (* k k))
   (if (<= m 1.1)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.165) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.165)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.165], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.165000000000000008

   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. lower-+.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 68.7%

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

   if -0.165000000000000008 < m < 1.1000000000000001

   1. Initial program 92.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 92.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.2

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

   4. Applied rewrites 92.2%

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

   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. lower-+.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 1.1000000000000001 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.165:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.165:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.165)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.165) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.165)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.165], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.165000000000000008

   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. lower-+.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 68.7%

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

   if -0.165000000000000008 < m < 1.1000000000000001

   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. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 1.1000000000000001 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

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

Alternative 8: 61.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0035:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.0035)
   (/ a (* k k))
   (if (<= m 1.1) (/ a (fma 10.0 k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0035) {
		tmp = a / (k * k);
	} else if (m <= 1.1) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.0035)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.1)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.0035], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.00350000000000000007

   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. lower-+.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 68.7%

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

   if -0.00350000000000000007 < m < 1.1000000000000001

   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. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.6%

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

   if 1.1000000000000001 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

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

Alternative 9: 55.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.1) (/ a (* k k)) (* (* (* k k) a) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.1) {
		tmp = a / (k * k);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	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 <= 1.1d0) then
        tmp = a / (k * k)
    else
        tmp = ((k * k) * a) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.1) {
		tmp = a / (k * k);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.1:
		tmp = a / (k * k)
	else:
		tmp = ((k * k) * a) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.1)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.1)
		tmp = a / (k * k);
	else
		tmp = ((k * k) * a) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.1], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.1000000000000001

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.2%

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

   if 1.1000000000000001 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

       \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.28999999999999998

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.5%

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

   10. Applied rewrites 23.5%

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

   if 0.28999999999999998 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

      \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.4%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 56.6%

       \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.28999999999999998

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.5%

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

   10. Applied rewrites 23.5%

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

   if 0.28999999999999998 < m

   1. Initial program 78.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 48.9%

       \[\leadsto \left(\left(a \cdot k\right) \cdot k\right) \cdot 99 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 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 89.1%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. lower-+.f64 42.3

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

5. Applied rewrites 42.3%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 16.3%

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

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 9.5%

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

13. Applied rewrites 9.5%

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

Reproduce

herbie shell --seed 2024359 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))