Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Percentage Accurate: 91.1% → 95.3%

Time: 4.3s

Alternatives: 12

Speedup: 0.7×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m + a\_m} - \left(9 \cdot z\right) \cdot \frac{t}{a\_m + a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 1.45e-37)
    (/ (fma (* 0.5 y) x (* (* -4.5 t) z)) a_m)
    (- (* y (/ x (+ a_m a_m))) (* (* 9.0 z) (/ t (+ a_m a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1.45e-37) {
		tmp = fma((0.5 * y), x, ((-4.5 * t) * z)) / a_m;
	} else {
		tmp = (y * (x / (a_m + a_m))) - ((9.0 * z) * (t / (a_m + a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1.45e-37)
		tmp = Float64(fma(Float64(0.5 * y), x, Float64(Float64(-4.5 * t) * z)) / a_m);
	else
		tmp = Float64(Float64(y * Float64(x / Float64(a_m + a_m))) - Float64(Float64(9.0 * z) * Float64(t / Float64(a_m + a_m))));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1.45e-37], N[(N[(N[(0.5 * y), $MachinePrecision] * x + N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y * N[(x / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(9.0 * z), $MachinePrecision] * N[(t / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.45000000000000002e-37

   1. Initial program 97.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(y \cdot x\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot y\right) \cdot x + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      12. lower-*.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a}} \]

   if 1.45000000000000002e-37 < a

   1. Initial program 85.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      10. lower-/.f64 N/A

          \[\leadsto y \cdot \color{blue}{\frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      11. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{2 \cdot a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      12. count-2-rev N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{a + a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      13. lower-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{a + a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      15. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      16. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(9 \cdot z\right)} \cdot t}{a \cdot 2} \]

      17. associate-/l* N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right) \cdot \frac{t}{a \cdot 2}} \]

      18. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right) \cdot \frac{t}{a \cdot 2}} \]

      19. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right)} \cdot \frac{t}{a \cdot 2} \]

      20. lower-/.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \color{blue}{\frac{t}{a \cdot 2}} \]

      21. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{2 \cdot a}} \]

      22. count-2-rev N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{a + a}} \]

      23. lower-+.f64 92.3

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{a + a}} \]

   3. Applied rewrites 92.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a\_m \cdot x}, \frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (fma (* -4.5 t) (/ z (* a_m x)) (* (/ y a_m) 0.5)) x))
        (t_2 (- (* x y) (* (* z 9.0) t))))
   (*
    a_s
    (if (<= t_2 -2e+290)
      t_1
      (if (<= t_2 5e+284) (/ (fma (* 0.5 y) x (* (* -4.5 t) z)) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = fma((-4.5 * t), (z / (a_m * x)), ((y / a_m) * 0.5)) * x;
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -2e+290) {
		tmp = t_1;
	} else if (t_2 <= 5e+284) {
		tmp = fma((0.5 * y), x, ((-4.5 * t) * z)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(fma(Float64(-4.5 * t), Float64(z / Float64(a_m * x)), Float64(Float64(y / a_m) * 0.5)) * x)
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= -2e+290)
		tmp = t_1;
	elseif (t_2 <= 5e+284)
		tmp = Float64(fma(Float64(0.5 * y), x, Float64(Float64(-4.5 * t) * z)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(-4.5 * t), $MachinePrecision] * N[(z / N[(a$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -2e+290], t$95$1, If[LessEqual[t$95$2, 5e+284], N[(N[(N[(0.5 * y), $MachinePrecision] * x + N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -2.00000000000000012e290 or 4.9999999999999999e284 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 70.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 85.7

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   if -2.00000000000000012e290 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 4.9999999999999999e284

   1. Initial program 98.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(y \cdot x\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot y\right) \cdot x + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      12. lower-*.f64 98.8

          \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) (- INFINITY))
      t_1
      (if (<= (* x y) 5e+251)
        (/ (fma (* 0.5 y) x (* (* -4.5 t) z)) a_m)
        t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= 5e+251) {
		tmp = fma((0.5 * y), x, ((-4.5 * t) * z)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+251)
		tmp = Float64(fma(Float64(0.5 * y), x, Float64(Float64(-4.5 * t) * z)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+251], N[(N[(N[(0.5 * y), $MachinePrecision] * x + N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 5.0000000000000005e251 < (*.f64 x y)

   1. Initial program 70.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 95.7

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 94.4

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 94.4%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -inf.0 < (*.f64 x y) < 5.0000000000000005e251

   1. Initial program 94.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(y \cdot x\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot y\right) \cdot x + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      12. lower-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) (- INFINITY))
      t_1
      (if (<= (* x y) 5e+251)
        (/ (fma (* -9.0 z) t (* y x)) (+ a_m a_m))
        t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= 5e+251) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+251)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+251], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 5.0000000000000005e251 < (*.f64 x y)

   1. Initial program 70.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 95.7

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 94.4

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 94.4%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -inf.0 < (*.f64 x y) < 5.0000000000000005e251

   1. Initial program 94.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 94.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 94.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 35000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a\_m + a\_m}, \frac{-9 \cdot \left(t \cdot z\right)}{a\_m + a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 35000000000000.0)
    (/ (fma (* 0.5 y) x (* (* -4.5 t) z)) a_m)
    (fma y (/ x (+ a_m a_m)) (/ (* -9.0 (* t z)) (+ a_m a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 35000000000000.0) {
		tmp = fma((0.5 * y), x, ((-4.5 * t) * z)) / a_m;
	} else {
		tmp = fma(y, (x / (a_m + a_m)), ((-9.0 * (t * z)) / (a_m + a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 35000000000000.0)
		tmp = Float64(fma(Float64(0.5 * y), x, Float64(Float64(-4.5 * t) * z)) / a_m);
	else
		tmp = fma(y, Float64(x / Float64(a_m + a_m)), Float64(Float64(-9.0 * Float64(t * z)) / Float64(a_m + a_m)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 35000000000000.0], N[(N[(N[(0.5 * y), $MachinePrecision] * x + N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e13

   1. Initial program 97.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(y \cdot x\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot y\right) \cdot x + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      12. lower-*.f64 98.2

          \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a} \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a}} \]

   if 3.5e13 < a

   1. Initial program 83.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} + \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} + \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{a \cdot 2}, \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{x}{a \cdot 2}}, \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{2 \cdot a}}, \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}\right) \]

      13. count-2-rev N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{a + a}}, \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{a + a}}, \frac{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t}{a \cdot 2}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t}{a \cdot 2}\right) \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t}{a \cdot 2}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t}{a \cdot 2}\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{\color{blue}{-9 \cdot \left(z \cdot t\right)}}{a \cdot 2}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{a + a}, \frac{-9 \cdot \color{blue}{\left(t \cdot z\right)}}{a \cdot 2}\right) \]

   3. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{a + a}, \frac{-9 \cdot \left(t \cdot z\right)}{a + a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot y, \frac{x}{a\_m}, \frac{t \cdot z}{a\_m} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 1e-37)
    (/ (fma (* 0.5 y) x (* (* -4.5 t) z)) a_m)
    (fma (* 0.5 y) (/ x a_m) (* (/ (* t z) a_m) -4.5)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1e-37) {
		tmp = fma((0.5 * y), x, ((-4.5 * t) * z)) / a_m;
	} else {
		tmp = fma((0.5 * y), (x / a_m), (((t * z) / a_m) * -4.5));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1e-37)
		tmp = Float64(fma(Float64(0.5 * y), x, Float64(Float64(-4.5 * t) * z)) / a_m);
	else
		tmp = fma(Float64(0.5 * y), Float64(x / a_m), Float64(Float64(Float64(t * z) / a_m) * -4.5));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1e-37], N[(N[(N[(0.5 * y), $MachinePrecision] * x + N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision] + N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.00000000000000007e-37

   1. Initial program 97.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{x \cdot y}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a} + \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot y\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(y \cdot x\right) + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot y\right) \cdot x + \frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \frac{-9}{2} \cdot \left(t \cdot z\right)\right)}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y, x, \left(\frac{-9}{2} \cdot t\right) \cdot z\right)}{a} \]

      12. lower-*.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot y, x, \left(-4.5 \cdot t\right) \cdot z\right)}{a}} \]

   if 1.00000000000000007e-37 < a

   1. Initial program 85.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 86.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 86.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, y \cdot x\right)}{a + a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a + a} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-9 \cdot z\right) \cdot t + y \cdot x}}{a + a} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x + \left(-9 \cdot z\right) \cdot t}}{a + a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} + \left(-9 \cdot z\right) \cdot t}{a + a} \]

      8. associate-*l* N/A

         \[\leadsto \frac{x \cdot y + \color{blue}{-9 \cdot \left(z \cdot t\right)}}{a + a} \]

      9. *-commutative N/A

         \[\leadsto \frac{x \cdot y + -9 \cdot \color{blue}{\left(t \cdot z\right)}}{a + a} \]

      10. div-add N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{a + a} + \frac{-9 \cdot \left(t \cdot z\right)}{a + a}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot x}}{a + a} + \frac{-9 \cdot \left(t \cdot z\right)}{a + a} \]

      12. count-2-rev N/A

          \[\leadsto \frac{y \cdot x}{\color{blue}{2 \cdot a}} + \frac{-9 \cdot \left(t \cdot z\right)}{a + a} \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\frac{y}{2} \cdot \frac{x}{a}} + \frac{-9 \cdot \left(t \cdot z\right)}{a + a} \]

      14. count-2-rev N/A

          \[\leadsto \frac{y}{2} \cdot \frac{x}{a} + \frac{-9 \cdot \left(t \cdot z\right)}{\color{blue}{2 \cdot a}} \]

      15. times-frac N/A

          \[\leadsto \frac{y}{2} \cdot \frac{x}{a} + \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{y}{2} \cdot \frac{x}{a} + \color{blue}{\frac{-9}{2}} \cdot \frac{t \cdot z}{a} \]

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{2}, \frac{x}{a}, \frac{-9}{2} \cdot \frac{t \cdot z}{a}\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{2}}, \frac{x}{a}, \frac{-9}{2} \cdot \frac{t \cdot z}{a}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{2}, \color{blue}{\frac{x}{a}}, \frac{-9}{2} \cdot \frac{t \cdot z}{a}\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{2}, \frac{x}{a}, \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}}\right) \]

      21. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{2}, \frac{x}{a}, \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{2}, \frac{x}{a}, \frac{\color{blue}{t \cdot z}}{a} \cdot \frac{-9}{2}\right) \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{2}, \frac{x}{a}, \frac{t \cdot z}{a} \cdot -4.5\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot y}, \frac{x}{a}, \frac{t \cdot z}{a} \cdot \frac{-9}{2}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 89.2

         \[\leadsto \mathsf{fma}\left(0.5 \cdot \color{blue}{y}, \frac{x}{a}, \frac{t \cdot z}{a} \cdot -4.5\right) \]

   8. Applied rewrites 89.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 \cdot y}, \frac{x}{a}, \frac{t \cdot z}{a} \cdot -4.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{t \cdot -9}{a\_m} \cdot \frac{z}{2}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) -2e+34)
      t_1
      (if (<= (* x y) 4e-25)
        (* (/ (* t -9.0) a_m) (/ z 2.0))
        (if (<= (* x y) 5e+156) (/ (* x y) (+ a_m a_m)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((t * -9.0) / a_m) * (z / 2.0);
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a_m) * 0.5d0) * x
    if ((x * y) <= (-2d+34)) then
        tmp = t_1
    else if ((x * y) <= 4d-25) then
        tmp = ((t * (-9.0d0)) / a_m) * (z / 2.0d0)
    else if ((x * y) <= 5d+156) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((t * -9.0) / a_m) * (z / 2.0);
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	tmp = 0
	if (x * y) <= -2e+34:
		tmp = t_1
	elif (x * y) <= 4e-25:
		tmp = ((t * -9.0) / a_m) * (z / 2.0)
	elif (x * y) <= 5e+156:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= -2e+34)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-25)
		tmp = Float64(Float64(Float64(t * -9.0) / a_m) * Float64(z / 2.0));
	elseif (Float64(x * y) <= 5e+156)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	tmp = 0.0;
	if ((x * y) <= -2e+34)
		tmp = t_1;
	elseif ((x * y) <= 4e-25)
		tmp = ((t * -9.0) / a_m) * (z / 2.0);
	elseif ((x * y) <= 5e+156)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+34], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-25], N[(N[(N[(t * -9.0), $MachinePrecision] / a$95$m), $MachinePrecision] * N[(z / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+156], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999989e34 or 4.99999999999999992e156 < (*.f64 x y)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 89.4

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 82.1

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 82.1%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -1.99999999999999989e34 < (*.f64 x y) < 4.00000000000000015e-25

   1. Initial program 94.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 75.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. metadata-eval N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{\color{blue}{2}} \]

      5. times-frac N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot -9}{\color{blue}{a \cdot 2}} \]

      6. *-commutative N/A

         \[\leadsto \frac{-9 \cdot \left(t \cdot z\right)}{\color{blue}{a} \cdot 2} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(-9 \cdot t\right) \cdot z}{\color{blue}{a} \cdot 2} \]

      8. times-frac N/A

         \[\leadsto \frac{-9 \cdot t}{a} \cdot \color{blue}{\frac{z}{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-9 \cdot t}{a} \cdot \color{blue}{\frac{z}{2}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{-9 \cdot t}{a} \cdot \frac{\color{blue}{z}}{2} \]

      11. *-commutative N/A

          \[\leadsto \frac{t \cdot -9}{a} \cdot \frac{z}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{t \cdot -9}{a} \cdot \frac{z}{2} \]

      13. lower-/.f64 73.4

          \[\leadsto \frac{t \cdot -9}{a} \cdot \frac{z}{\color{blue}{2}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \frac{t \cdot -9}{a} \cdot \color{blue}{\frac{z}{2}} \]

   if 4.00000000000000015e-25 < (*.f64 x y) < 4.99999999999999992e156

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
   5. Step-by-step derivation

      1. lower-*.f64 63.2

         \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

   6. Applied rewrites 63.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) -2e+34)
      t_1
      (if (<= (* x y) 4e-25)
        (/ (* (* -4.5 t) z) a_m)
        (if (<= (* x y) 5e+156) (/ (* x y) (+ a_m a_m)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a_m) * 0.5d0) * x
    if ((x * y) <= (-2d+34)) then
        tmp = t_1
    else if ((x * y) <= 4d-25) then
        tmp = (((-4.5d0) * t) * z) / a_m
    else if ((x * y) <= 5d+156) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	tmp = 0
	if (x * y) <= -2e+34:
		tmp = t_1
	elif (x * y) <= 4e-25:
		tmp = ((-4.5 * t) * z) / a_m
	elif (x * y) <= 5e+156:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= -2e+34)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-25)
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a_m);
	elseif (Float64(x * y) <= 5e+156)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	tmp = 0.0;
	if ((x * y) <= -2e+34)
		tmp = t_1;
	elseif ((x * y) <= 4e-25)
		tmp = ((-4.5 * t) * z) / a_m;
	elseif ((x * y) <= 5e+156)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+34], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-25], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+156], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999989e34 or 4.99999999999999992e156 < (*.f64 x y)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 89.4

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 82.1

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 82.1%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -1.99999999999999989e34 < (*.f64 x y) < 4.00000000000000015e-25

   1. Initial program 94.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 75.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      9. lower-/.f64 75.2

         \[\leadsto \frac{\left(-4.5 \cdot t\right) \cdot z}{\color{blue}{a}} \]

   6. Applied rewrites 75.2%

      \[\leadsto \frac{\left(-4.5 \cdot t\right) \cdot z}{\color{blue}{a}} \]

   if 4.00000000000000015e-25 < (*.f64 x y) < 4.99999999999999992e156

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
   5. Step-by-step derivation

      1. lower-*.f64 63.2

         \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

   6. Applied rewrites 63.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{t \cdot z}{a\_m} \cdot -4.5\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) -2e+34)
      t_1
      (if (<= (* x y) 4e-25)
        (* (/ (* t z) a_m) -4.5)
        (if (<= (* x y) 5e+156) (/ (* x y) (+ a_m a_m)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((t * z) / a_m) * -4.5;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a_m) * 0.5d0) * x
    if ((x * y) <= (-2d+34)) then
        tmp = t_1
    else if ((x * y) <= 4d-25) then
        tmp = ((t * z) / a_m) * (-4.5d0)
    else if ((x * y) <= 5d+156) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((t * z) / a_m) * -4.5;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	tmp = 0
	if (x * y) <= -2e+34:
		tmp = t_1
	elif (x * y) <= 4e-25:
		tmp = ((t * z) / a_m) * -4.5
	elif (x * y) <= 5e+156:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= -2e+34)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-25)
		tmp = Float64(Float64(Float64(t * z) / a_m) * -4.5);
	elseif (Float64(x * y) <= 5e+156)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	tmp = 0.0;
	if ((x * y) <= -2e+34)
		tmp = t_1;
	elseif ((x * y) <= 4e-25)
		tmp = ((t * z) / a_m) * -4.5;
	elseif ((x * y) <= 5e+156)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+34], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-25], N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+156], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999989e34 or 4.99999999999999992e156 < (*.f64 x y)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 89.4

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 82.1

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 82.1%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -1.99999999999999989e34 < (*.f64 x y) < 4.00000000000000015e-25

   1. Initial program 94.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 75.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   if 4.00000000000000015e-25 < (*.f64 x y) < 4.99999999999999992e156

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
   5. Step-by-step derivation

      1. lower-*.f64 63.2

         \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

   6. Applied rewrites 63.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{z}{a\_m} \cdot -4.5\right) \cdot t\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) -2e+34)
      t_1
      (if (<= (* x y) 4e-25)
        (* (* (/ z a_m) -4.5) t)
        (if (<= (* x y) 5e+156) (/ (* x y) (+ a_m a_m)) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((z / a_m) * -4.5) * t;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a_m) * 0.5d0) * x
    if ((x * y) <= (-2d+34)) then
        tmp = t_1
    else if ((x * y) <= 4d-25) then
        tmp = ((z / a_m) * (-4.5d0)) * t
    else if ((x * y) <= 5d+156) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -2e+34) {
		tmp = t_1;
	} else if ((x * y) <= 4e-25) {
		tmp = ((z / a_m) * -4.5) * t;
	} else if ((x * y) <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	tmp = 0
	if (x * y) <= -2e+34:
		tmp = t_1
	elif (x * y) <= 4e-25:
		tmp = ((z / a_m) * -4.5) * t
	elif (x * y) <= 5e+156:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= -2e+34)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-25)
		tmp = Float64(Float64(Float64(z / a_m) * -4.5) * t);
	elseif (Float64(x * y) <= 5e+156)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	tmp = 0.0;
	if ((x * y) <= -2e+34)
		tmp = t_1;
	elseif ((x * y) <= 4e-25)
		tmp = ((z / a_m) * -4.5) * t;
	elseif ((x * y) <= 5e+156)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+34], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-25], N[(N[(N[(z / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+156], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999989e34 or 4.99999999999999992e156 < (*.f64 x y)

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 89.4

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 82.1

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 82.1%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -1.99999999999999989e34 < (*.f64 x y) < 4.00000000000000015e-25

   1. Initial program 94.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 75.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{a} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-9}{2} \cdot t\right) \cdot z}{a} \]

      9. lower-/.f64 75.2

         \[\leadsto \frac{\left(-4.5 \cdot t\right) \cdot z}{\color{blue}{a}} \]

   6. Applied rewrites 75.2%

      \[\leadsto \frac{\left(-4.5 \cdot t\right) \cdot z}{\color{blue}{a}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot \color{blue}{t} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot t} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot t}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot t}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot t}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot t}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, \frac{1}{2}, \frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, \frac{1}{2}, \frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

      13. lower-/.f64 81.9

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, 0.5, \frac{z}{a} \cdot -4.5\right) \cdot t \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{t \cdot a}, 0.5, \frac{z}{a} \cdot -4.5\right) \cdot t} \]

   10. Taylor expanded in x around 0

       \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

       3. lift-*.f64 73.6

          \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]

   12. Applied rewrites 73.6%

       \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]

   if 4.00000000000000015e-25 < (*.f64 x y) < 4.99999999999999992e156

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.4

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
   5. Step-by-step derivation

      1. lower-*.f64 63.2

         \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

   6. Applied rewrites 63.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)) (t_2 (- (* x y) (* (* z 9.0) t))))
   (*
    a_s
    (if (<= t_2 -1e+94)
      t_1
      (if (<= t_2 5e+156) (/ (* x y) (+ a_m a_m)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -1e+94) {
		tmp = t_1;
	} else if (t_2 <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / a_m) * 0.5d0) * x
    t_2 = (x * y) - ((z * 9.0d0) * t)
    if (t_2 <= (-1d+94)) then
        tmp = t_1
    else if (t_2 <= 5d+156) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -1e+94) {
		tmp = t_1;
	} else if (t_2 <= 5e+156) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	t_2 = (x * y) - ((z * 9.0) * t)
	tmp = 0
	if t_2 <= -1e+94:
		tmp = t_1
	elif t_2 <= 5e+156:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= -1e+94)
		tmp = t_1;
	elseif (t_2 <= 5e+156)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	t_2 = (x * y) - ((z * 9.0) * t);
	tmp = 0.0;
	if (t_2 <= -1e+94)
		tmp = t_1;
	elseif (t_2 <= 5e+156)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -1e+94], t$95$1, If[LessEqual[t$95$2, 5e+156], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -1e94 or 4.99999999999999992e156 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 85.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot \color{blue}{x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \left(t \cdot \frac{z}{a \cdot x}\right) + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-9}{2} \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      11. lower-/.f64 83.3

          \[\leadsto \mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x \]

   4. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5 \cdot t, \frac{z}{a \cdot x}, \frac{y}{a} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \frac{y}{a}\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{y}{a} \cdot \frac{1}{2}\right) \cdot x \]

      3. lift-*.f64 53.6

         \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 53.6%

      \[\leadsto \left(\frac{y}{a} \cdot 0.5\right) \cdot x \]

   if -1e94 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 4.99999999999999992e156

   1. Initial program 98.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 98.3

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 98.3

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
   5. Step-by-step derivation

      1. lower-*.f64 56.3

         \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

   6. Applied rewrites 56.3%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \frac{x \cdot y}{a\_m + a\_m} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (/ (* x y) (+ a_m a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / (a_m + a_m));
}

Fortran

a\_m =     private
a\_s =     private
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * ((x * y) / (a_m + a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / (a_m + a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((x * y) / (a_m + a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(x * y) / Float64(a_m + a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((x * y) / (a_m + a_m));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

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

      \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   13. lower-*.f64 91.4

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

   15. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

   16. count-2-rev N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   17. lower-+.f64 91.4

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

3. Applied rewrites 91.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

4. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
5. Step-by-step derivation

   1. lower-*.f64 51.1

      \[\leadsto \frac{x \cdot \color{blue}{y}}{a + a} \]

6. Applied rewrites 51.1%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64
  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))