Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.6%

Time: 5.4s

Alternatives: 12

Speedup: 1.1×

• 22.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \frac{1}{{k}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.85)
   (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (/ 1.0 (pow k -2.0))))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.85) {
		tmp = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (1.0 / pow(k, -2.0)));
	} 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 <= 2.85d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (1.0d0 / (k ** (-2.0d0))))
    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 <= 2.85) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (1.0 / Math.pow(k, -2.0)));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.85:
		tmp = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (1.0 / math.pow(k, -2.0)))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.85)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(1.0 / (k ^ -2.0))));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.85)
		tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (1.0 / (k ^ -2.0)));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.85], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.85000000000000009

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. pow-to-exp N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. pow2 N/A

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

      11. inv-pow N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. pow2 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. pow-flip N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 98.2

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

   6. Applied rewrites 98.2%

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

   if 2.85000000000000009 < m

   1. Initial program 79.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.85)
   (/ (* 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 <= 2.85) {
		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 <= 2.85d0) 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 <= 2.85) {
		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 <= 2.85:
		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 <= 2.85)
		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 <= 2.85)
		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, 2.85], 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 < 2.85000000000000009

   1. Initial program 98.2%

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

   if 2.85000000000000009 < m

   1. Initial program 79.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-17} \lor \neg \left(m \leq 0.0037\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2.05e-17) (not (<= m 0.0037)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.05e-17) || !(m <= 0.0037)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2.05e-17) || !(m <= 0.0037))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2.05e-17], N[Not[LessEqual[m, 0.0037]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.05e-17 or 0.0037000000000000002 < m

   1. Initial program 89.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.05e-17 < m < 0.0037000000000000002

   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

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-17} \lor \neg \left(m \leq 0.0037\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(10 + k\right)\\ \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 22500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{\frac{1 - t\_0 \cdot t\_0}{\left(-k\right) \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ 10.0 k))))
   (if (<= m -0.49)
     (/ (* (/ a (* k k)) 99.0) (* k k))
     (if (<= m 22500000.0)
       (/ a (fma (+ 10.0 k) k 1.0))
       (/ (* a 1.0) (/ (- 1.0 (* t_0 t_0)) (* (- k) k)))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (10.0 + k);
	double tmp;
	if (m <= -0.49) {
		tmp = ((a / (k * k)) * 99.0) / (k * k);
	} else if (m <= 22500000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (a * 1.0) / ((1.0 - (t_0 * t_0)) / (-k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(10.0 + k))
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(Float64(Float64(a / Float64(k * k)) * 99.0) / Float64(k * k));
	elseif (m <= 22500000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(a * 1.0) / Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(Float64(-k) * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.49], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 22500000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 1.0), $MachinePrecision] / N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-k) * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 60.8

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -0.48999999999999999 < m < 2.25e7

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 2.25e7 < m

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-+r+ N/A

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

      7. pow2 N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. flip-- N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 44.2

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

   10. Applied rewrites 44.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 22500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot 1}{\frac{1 - \left(k \cdot \left(10 + k\right)\right) \cdot \left(k \cdot \left(10 + k\right)\right)}{\left(-k\right) \cdot k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{\frac{a}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;m \leq 4.7 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.49)
   (/ (* (/ a (* k k)) 99.0) (* k k))
   (if (<= m 1.85)
     (/ a (fma (+ 10.0 k) k 1.0))
     (if (<= m 4.7e+123)
       (fma (fma (* 99.0 a) k (* -10.0 a)) k a)
       (* (* -10.0 a) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.49) {
		tmp = ((a / (k * k)) * 99.0) / (k * k);
	} else if (m <= 1.85) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (m <= 4.7e+123) {
		tmp = fma(fma((99.0 * a), k, (-10.0 * a)), k, a);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(Float64(Float64(a / Float64(k * k)) * 99.0) / Float64(k * k));
	elseif (m <= 1.85)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (m <= 4.7e+123)
		tmp = fma(fma(Float64(99.0 * a), k, Float64(-10.0 * a)), k, a);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.49], N[(N[(N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.7e+123], N[(N[(N[(99.0 * a), $MachinePrecision] * k + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 60.8

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -0.48999999999999999 < m < 1.8500000000000001

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 1.8500000000000001 < m < 4.69999999999999979e123

   1. Initial program 74.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 15.0%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 4.69999999999999979e123 < m

   1. Initial program 84.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 28.3

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

   11. Applied rewrites 28.3%

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

Alternative 6: 58.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.85:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;m \leq 4.7 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99 \cdot a, k, -10 \cdot a\right), k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.49)
   (/ a (* k k))
   (if (<= m 1.85)
     (/ a (fma (+ 10.0 k) k 1.0))
     (if (<= m 4.7e+123)
       (fma (fma (* 99.0 a) k (* -10.0 a)) k a)
       (* (* -10.0 a) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.49) {
		tmp = a / (k * k);
	} else if (m <= 1.85) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (m <= 4.7e+123) {
		tmp = fma(fma((99.0 * a), k, (-10.0 * a)), k, a);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.85)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (m <= 4.7e+123)
		tmp = fma(fma(Float64(99.0 * a), k, Float64(-10.0 * a)), k, a);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.49], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.7e+123], N[(N[(N[(99.0 * a), $MachinePrecision] * k + N[(-10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.9%

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

   if -0.48999999999999999 < m < 1.8500000000000001

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 1.8500000000000001 < m < 4.69999999999999979e123

   1. Initial program 74.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 15.0%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 28.6

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

   11. Applied rewrites 28.6%

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

   if 4.69999999999999979e123 < m

   1. Initial program 84.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 28.3

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

   11. Applied rewrites 28.3%

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

Alternative 7: 45.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -8.8 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-306}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k -8.8e-103)
     t_0
     (if (<= k 2.65e-306)
       (* (* -10.0 a) k)
       (if (<= k 0.1) (fma (* -10.0 a) k a) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= -8.8e-103) {
		tmp = t_0;
	} else if (k <= 2.65e-306) {
		tmp = (-10.0 * a) * k;
	} else if (k <= 0.1) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= -8.8e-103)
		tmp = t_0;
	elseif (k <= 2.65e-306)
		tmp = Float64(Float64(-10.0 * a) * k);
	elseif (k <= 0.1)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -8.8e-103], t$95$0, If[LessEqual[k, 2.65e-306], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision], If[LessEqual[k, 0.1], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if k < -8.7999999999999997e-103 or 0.10000000000000001 < k

   1. Initial program 83.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 51.7%

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

   if -8.7999999999999997e-103 < k < 2.6499999999999999e-306

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 4.4

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 50.5

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

   11. Applied rewrites 50.5%

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

   if 2.6499999999999999e-306 < k < 0.10000000000000001

   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

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 51.0

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

   10. Applied rewrites 51.0%

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

Alternative 8: 57.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 22500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.49)
   (/ a (* k k))
   (if (<= m 22500000.0) (/ a (fma (+ 10.0 k) k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.49) {
		tmp = a / (k * k);
	} else if (m <= 22500000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 22500000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.49], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 22500000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.9%

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

   if -0.48999999999999999 < m < 2.25e7

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 2.25e7 < m

   1. Initial program 82.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 24.5

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

   11. Applied rewrites 24.5%

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

Alternative 9: 56.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 22500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.49)
   (/ a (* k k))
   (if (<= m 22500000.0) (/ a (fma k k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.49) {
		tmp = a / (k * k);
	} else if (m <= 22500000.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 22500000.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.49], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 22500000.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.9%

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

   if -0.48999999999999999 < m < 2.25e7

   1. Initial program 93.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 89.5%

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

   if 2.25e7 < m

   1. Initial program 82.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 24.5

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

   11. Applied rewrites 24.5%

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

Alternative 10: 47.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 22500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.25e-38)
   (/ a (* k k))
   (if (<= m 22500000.0) (/ a (fma 10.0 k 1.0)) (* (* -10.0 a) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.25e-38) {
		tmp = a / (k * k);
	} else if (m <= 22500000.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.25e-38)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 22500000.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.25e-38], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 22500000.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.25000000000000008e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 38.1

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

   5. Applied rewrites 38.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.6%

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

   if -1.25000000000000008e-38 < m < 2.25e7

   1. Initial program 93.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   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 68.7%

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

   if 2.25e7 < m

   1. Initial program 82.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 24.5

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

   11. Applied rewrites 24.5%

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

Alternative 11: 25.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 22500000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 22500000.0) a (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 22500000.0) {
		tmp = a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	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 <= 22500000.0d0) then
        tmp = a
    else
        tmp = ((-10.0d0) * a) * k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 22500000.0) {
		tmp = a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 22500000.0:
		tmp = a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 22500000.0)
		tmp = a;
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 22500000.0)
		tmp = a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 22500000.0], a, N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.25e7

   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

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 31.7%

      \[\leadsto a \]

   if 2.25e7 < m

   1. Initial program 82.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 3.3

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 24.5

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

   11. Applied rewrites 24.5%

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

Alternative 12: 19.9% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 46.1

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

5. Applied rewrites 46.1%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 22.4%

   \[\leadsto a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025084 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))