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

Percentage Accurate: 91.6% → 97.3%

Time: 7.8s

Alternatives: 12

Speedup: 0.5×

• 29.3% 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: 91.6% 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: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+286}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -1e+298) (not (<= t_1 2e+286)))
     (fma (/ (/ y a) 2.0) x (* (- t) (* 4.5 (/ z a))))
     (/ (fma (* t z) -9.0 (* y x)) (* a 2.0)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -1e+298) || !(t_1 <= 2e+286)) {
		tmp = fma(((y / a) / 2.0), x, (-t * (4.5 * (z / a))));
	} else {
		tmp = fma((t * z), -9.0, (y * x)) / (a * 2.0);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -1e+298) || !(t_1 <= 2e+286))
		tmp = fma(Float64(Float64(y / a) / 2.0), x, Float64(Float64(-t) * Float64(4.5 * Float64(z / a))));
	else
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+298], N[Not[LessEqual[t$95$1, 2e+286]], $MachinePrecision]], N[(N[(N[(y / a), $MachinePrecision] / 2.0), $MachinePrecision] * x + N[((-t) * N[(4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999996e297 or 2.00000000000000007e286 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 96.9%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2.00000000000000007e286

   1. Initial program 99.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 99.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+298} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+286}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t} \cdot 0.5, -4.5 \cdot z\right)}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a) x)
     (if (<= t_1 5e+281)
       (/ (fma (* t z) -9.0 (* y x)) (* a 2.0))
       (* (/ (fma x (* (/ y t) 0.5) (* -4.5 z)) a) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a) * x;
	} else if (t_1 <= 5e+281) {
		tmp = fma((t * z), -9.0, (y * x)) / (a * 2.0);
	} else {
		tmp = (fma(x, ((y / t) * 0.5), (-4.5 * z)) / a) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a) * x);
	elseif (t_1 <= 5e+281)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(x, Float64(Float64(y / t) * 0.5), Float64(-4.5 * z)) / a) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e+281], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision] + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 75.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.2%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 5.00000000000000016e281

   1. Initial program 99.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 5.00000000000000016e281 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 90.5%

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

Alternative 3: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298} \lor \neg \left(t\_1 \leq 10^{+290}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t} \cdot 0.5, -4.5 \cdot z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -1e+298) (not (<= t_1 1e+290)))
     (* (/ (fma x (* (/ y t) 0.5) (* -4.5 z)) a) t)
     (/ (fma (* t z) -9.0 (* y x)) (* a 2.0)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -1e+298) || !(t_1 <= 1e+290)) {
		tmp = (fma(x, ((y / t) * 0.5), (-4.5 * z)) / a) * t;
	} else {
		tmp = fma((t * z), -9.0, (y * x)) / (a * 2.0);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -1e+298) || !(t_1 <= 1e+290))
		tmp = Float64(Float64(fma(x, Float64(Float64(y / t) * 0.5), Float64(-4.5 * z)) / a) * t);
	else
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+298], N[Not[LessEqual[t$95$1, 1e+290]], $MachinePrecision]], N[(N[(N[(x * N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision] + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999996e297 or 1.00000000000000006e290 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 58.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 87.2%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.00000000000000006e290

   1. Initial program 99.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 99.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+298} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+290}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t} \cdot 0.5, -4.5 \cdot z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+272}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (/ (* -4.5 t) a) z)
     (if (<= t_1 1e+272)
       (/ (fma (* t z) -9.0 (* y x)) (* a 2.0))
       (* t (* -4.5 (/ z a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-4.5 * t) / a) * z;
	} else if (t_1 <= 1e+272) {
		tmp = fma((t * z), -9.0, (y * x)) / (a * 2.0);
	} else {
		tmp = t * (-4.5 * (z / a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-4.5 * t) / a) * z);
	elseif (t_1 <= 1e+272)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(-4.5 * Float64(z / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-4.5 * t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+272], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 42.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.9%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.0000000000000001e272

   1. Initial program 95.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 95.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if 1.0000000000000001e272 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 51.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 92.6%

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

Alternative 5: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+241}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (/ (* -4.5 t) a) z)
     (if (<= t_1 1e+241)
       (/ (fma y x (* (* -9.0 z) t)) (* a 2.0))
       (* t (* z (/ -4.5 a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-4.5 * t) / a) * z;
	} else if (t_1 <= 1e+241) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a * 2.0);
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-4.5 * t) / a) * z);
	elseif (t_1 <= 1e+241)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-4.5 * t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+241], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 42.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.9%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.0000000000000001e241

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow1 N/A

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

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. pow2 N/A

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

      18. sqrt-pow1 N/A

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

      19. metadata-eval N/A

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

      20. unpow1 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 N/A

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

      24. lower-*.f64 N/A

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

   4. Applied rewrites 95.4%

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

   if 1.0000000000000001e241 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 94.2%

      \[\leadsto t \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.1%

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

Alternative 6: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+46} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -1e+46) (not (<= t_1 2e+57)))
     (* t (* z (/ -4.5 a)))
     (/ (* y x) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (y * x) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-1d+46)) .or. (.not. (t_1 <= 2d+57))) then
        tmp = t * (z * ((-4.5d0) / a))
    else
        tmp = (y * x) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (y * x) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -1e+46) or not (t_1 <= 2e+57):
		tmp = t * (z * (-4.5 / a))
	else:
		tmp = (y * x) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57))
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	else
		tmp = Float64(Float64(y * x) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -1e+46) || ~((t_1 <= 2e+57)))
		tmp = t * (z * (-4.5 / a));
	else
		tmp = (y * x) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+46], N[Not[LessEqual[t$95$1, 2e+57]], $MachinePrecision]], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999999e45 or 2.0000000000000001e57 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 79.1%

      \[\leadsto t \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.0%

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

   if -9.9999999999999999e45 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000001e57

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

      2. lower-*.f64 71.7

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

   8. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 71.7

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

   10. Applied rewrites 71.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+46} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+46} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -1e+46) (not (<= t_1 2e+57)))
     (* t (* -4.5 (/ z a)))
     (/ (* y x) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57)) {
		tmp = t * (-4.5 * (z / a));
	} else {
		tmp = (y * x) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-1d+46)) .or. (.not. (t_1 <= 2d+57))) then
        tmp = t * ((-4.5d0) * (z / a))
    else
        tmp = (y * x) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57)) {
		tmp = t * (-4.5 * (z / a));
	} else {
		tmp = (y * x) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -1e+46) or not (t_1 <= 2e+57):
		tmp = t * (-4.5 * (z / a))
	else:
		tmp = (y * x) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -1e+46) || !(t_1 <= 2e+57))
		tmp = Float64(t * Float64(-4.5 * Float64(z / a)));
	else
		tmp = Float64(Float64(y * x) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -1e+46) || ~((t_1 <= 2e+57)))
		tmp = t * (-4.5 * (z / a));
	else
		tmp = (y * x) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+46], N[Not[LessEqual[t$95$1, 2e+57]], $MachinePrecision]], N[(t * N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999999e45 or 2.0000000000000001e57 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 79.1%

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

   if -9.9999999999999999e45 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000001e57

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

      2. lower-*.f64 71.7

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

   8. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 71.7

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

   10. Applied rewrites 71.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+46} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+57}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+17} \lor \neg \left(x \cdot y \leq 10^{-13}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* x y) -5e+17) (not (<= (* x y) 1e-13)))
   (* (* 0.5 x) (/ y a))
   (* t (* z (/ -4.5 a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -5e+17) || !((x * y) <= 1e-13)) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if (((x * y) <= (-5d+17)) .or. (.not. ((x * y) <= 1d-13))) then
        tmp = (0.5d0 * x) * (y / a)
    else
        tmp = t * (z * ((-4.5d0) / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -5e+17) || !((x * y) <= 1e-13)) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) <= -5e+17) or not ((x * y) <= 1e-13):
		tmp = (0.5 * x) * (y / a)
	else:
		tmp = t * (z * (-4.5 / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+17) || !(Float64(x * y) <= 1e-13))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	else
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) <= -5e+17) || ~(((x * y) <= 1e-13)))
		tmp = (0.5 * x) * (y / a);
	else
		tmp = t * (z * (-4.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+17], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-13]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e17 or 1e-13 < (*.f64 x y)

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

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

   if -5e17 < (*.f64 x y) < 1e-13

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 71.3%

      \[\leadsto t \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

      \[\leadsto t \cdot \color{blue}{\left(z \cdot \frac{-4.5}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+17} \lor \neg \left(x \cdot y \leq 10^{-13}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5 \cdot x}{a} \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-13}:\\ \;\;\;\;\frac{-4.5 \cdot \left(z \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+17)
   (* (/ (* 0.5 x) a) y)
   (if (<= (* x y) 1e-13) (/ (* -4.5 (* z t)) a) (* (* 0.5 x) (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = (-4.5 * (z * t)) / a;
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if ((x * y) <= (-5d+17)) then
        tmp = ((0.5d0 * x) / a) * y
    else if ((x * y) <= 1d-13) then
        tmp = ((-4.5d0) * (z * t)) / a
    else
        tmp = (0.5d0 * x) * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = (-4.5 * (z * t)) / a;
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+17:
		tmp = ((0.5 * x) / a) * y
	elif (x * y) <= 1e-13:
		tmp = (-4.5 * (z * t)) / a
	else:
		tmp = (0.5 * x) * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+17)
		tmp = Float64(Float64(Float64(0.5 * x) / a) * y);
	elseif (Float64(x * y) <= 1e-13)
		tmp = Float64(Float64(-4.5 * Float64(z * t)) / a);
	else
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+17)
		tmp = ((0.5 * x) / a) * y;
	elseif ((x * y) <= 1e-13)
		tmp = (-4.5 * (z * t)) / a;
	else
		tmp = (0.5 * x) * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+17], N[(N[(N[(0.5 * x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-13], N[(N[(-4.5 * N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e17

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]

   if -5e17 < (*.f64 x y) < 1e-13

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.1%

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

   if 1e-13 < (*.f64 x y)

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \frac{y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5 \cdot x}{a} \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-13}:\\ \;\;\;\;\frac{t \cdot z}{a} \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+17)
   (* (/ (* 0.5 x) a) y)
   (if (<= (* x y) 1e-13) (* (/ (* t z) a) -4.5) (* (* 0.5 x) (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = ((t * z) / a) * -4.5;
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if ((x * y) <= (-5d+17)) then
        tmp = ((0.5d0 * x) / a) * y
    else if ((x * y) <= 1d-13) then
        tmp = ((t * z) / a) * (-4.5d0)
    else
        tmp = (0.5d0 * x) * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = ((t * z) / a) * -4.5;
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+17:
		tmp = ((0.5 * x) / a) * y
	elif (x * y) <= 1e-13:
		tmp = ((t * z) / a) * -4.5
	else:
		tmp = (0.5 * x) * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+17)
		tmp = Float64(Float64(Float64(0.5 * x) / a) * y);
	elseif (Float64(x * y) <= 1e-13)
		tmp = Float64(Float64(Float64(t * z) / a) * -4.5);
	else
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+17)
		tmp = ((0.5 * x) / a) * y;
	elseif ((x * y) <= 1e-13)
		tmp = ((t * z) / a) * -4.5;
	else
		tmp = (0.5 * x) * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+17], N[(N[(N[(0.5 * x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-13], N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] * -4.5), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e17

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]

   if -5e17 < (*.f64 x y) < 1e-13

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 1e-13 < (*.f64 x y)

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \frac{y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{0.5 \cdot x}{a} \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-13}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+17)
   (* (/ (* 0.5 x) a) y)
   (if (<= (* x y) 1e-13) (* t (* z (/ -4.5 a))) (* (* 0.5 x) (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if ((x * y) <= (-5d+17)) then
        tmp = ((0.5d0 * x) / a) * y
    else if ((x * y) <= 1d-13) then
        tmp = t * (z * ((-4.5d0) / a))
    else
        tmp = (0.5d0 * x) * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+17) {
		tmp = ((0.5 * x) / a) * y;
	} else if ((x * y) <= 1e-13) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+17:
		tmp = ((0.5 * x) / a) * y
	elif (x * y) <= 1e-13:
		tmp = t * (z * (-4.5 / a))
	else:
		tmp = (0.5 * x) * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+17)
		tmp = Float64(Float64(Float64(0.5 * x) / a) * y);
	elseif (Float64(x * y) <= 1e-13)
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	else
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+17)
		tmp = ((0.5 * x) / a) * y;
	elseif ((x * y) <= 1e-13)
		tmp = t * (z * (-4.5 / a));
	else
		tmp = (0.5 * x) * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+17], N[(N[(N[(0.5 * x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-13], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e17

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]

   if -5e17 < (*.f64 x y) < 1e-13

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 71.3%

      \[\leadsto t \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.4%

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

   if 1e-13 < (*.f64 x y)

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot x}{a} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \frac{y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{y \cdot x}{a + a} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (/ (* y x) (+ a a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (y * x) / (a + a);
}

Fortran

NOTE: x, y, z, t, and a 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(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 = (y * x) / (a + a)
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (y * x) / (a + a);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (y * x) / (a + a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(y * x) / Float64(a + a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (y * x) / (a + a);
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(y * x), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

   2. lower-*.f64 53.6

      \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} \]

8. Applied rewrites 53.6%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot x}{\color{blue}{a \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y \cdot x}{\color{blue}{2 \cdot a}} \]

   3. count-2-rev N/A

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

   4. lower-+.f64 53.6

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

10. Applied rewrites 53.6%

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

Developer Target 1: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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
    real(8) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024360 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))