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

Percentage Accurate: 91.2% → 96.6%

Time: 4.0s

Alternatives: 6

Speedup: N/A×

• 28.4% 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: 91.2% 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: 96.6% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, -9 \cdot \left(\frac{z}{2} \cdot \frac{t}{a}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t \cdot -9}{2}, \frac{z}{a}, \frac{y}{2} \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= t_1 -5e+251)
     (fma (/ y a) (/ x 2.0) (* -9.0 (* (/ z 2.0) (/ t a))))
     (if (<= t_1 5e+220)
       (/ (fma (* -9.0 z) t (* y x)) (* a 2.0))
       (fma (/ (* t -9.0) 2.0) (/ z a) (* (/ y 2.0) (/ x a)))))))

C

assert(x < y && y < z && z < t && t < a);
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 <= -5e+251) {
		tmp = fma((y / a), (x / 2.0), (-9.0 * ((z / 2.0) * (t / a))));
	} else if (t_1 <= 5e+220) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a * 2.0);
	} else {
		tmp = fma(((t * -9.0) / 2.0), (z / a), ((y / 2.0) * (x / a)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
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 <= -5e+251)
		tmp = fma(Float64(y / a), Float64(x / 2.0), Float64(-9.0 * Float64(Float64(z / 2.0) * Float64(t / a))));
	elseif (t_1 <= 5e+220)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = fma(Float64(Float64(t * -9.0) / 2.0), Float64(z / a), Float64(Float64(y / 2.0) * Float64(x / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[LessEqual[t$95$1, -5e+251], N[(N[(y / a), $MachinePrecision] * N[(x / 2.0), $MachinePrecision] + N[(-9.0 * N[(N[(z / 2.0), $MachinePrecision] * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+220], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * -9.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[(z / a), $MachinePrecision] + N[(N[(y / 2.0), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -5.0000000000000005e251

   1. Initial program 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 94.1

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

   6. Applied rewrites 94.1%

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

   if -5.0000000000000005e251 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000002e220

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 5.0000000000000002e220 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 89.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. frac-times N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. metadata-eval N/A

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

      17. times-frac N/A

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

      18. associate-*r* N/A

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

      19. times-frac N/A

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

   6. Applied rewrites 91.4%

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

Alternative 2: 95.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, \frac{x}{2}, -9 \cdot \left(\frac{z}{2} \cdot \frac{t}{a}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= t_1 -5e+251)
     (fma (/ y a) (/ x 2.0) (* -9.0 (* (/ z 2.0) (/ t a))))
     (if (<= t_1 1e+285)
       (/ (fma (* -9.0 z) t (* y x)) (* a 2.0))
       (* (* (fma (/ x a) -0.5 (* (/ (/ (* t z) a) y) 4.5)) y) -1.0)))))

C

assert(x < y && y < z && z < t && t < a);
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 <= -5e+251) {
		tmp = fma((y / a), (x / 2.0), (-9.0 * ((z / 2.0) * (t / a))));
	} else if (t_1 <= 1e+285) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a * 2.0);
	} else {
		tmp = (fma((x / a), -0.5, ((((t * z) / a) / y) * 4.5)) * y) * -1.0;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
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 <= -5e+251)
		tmp = fma(Float64(y / a), Float64(x / 2.0), Float64(-9.0 * Float64(Float64(z / 2.0) * Float64(t / a))));
	elseif (t_1 <= 1e+285)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(Float64(x / a), -0.5, Float64(Float64(Float64(Float64(t * z) / a) / y) * 4.5)) * y) * -1.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[LessEqual[t$95$1, -5e+251], N[(N[(y / a), $MachinePrecision] * N[(x / 2.0), $MachinePrecision] + N[(-9.0 * N[(N[(z / 2.0), $MachinePrecision] * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+285], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / a), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -5.0000000000000005e251

   1. Initial program 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 94.1

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

   6. Applied rewrites 94.1%

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

   if -5.0000000000000005e251 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 9.9999999999999998e284

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 9.9999999999999998e284 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 66.0%

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

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 90.1

          \[\leadsto \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \]

   5. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.3% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+278}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (* (* (fma (/ x a) -0.5 (* (/ (/ (* t z) a) y) 4.5)) y) -1.0)))
   (if (<= (* x y) (- INFINITY))
     t_1
     (if (<= (* x y) 4e+278) (/ (fma (* -9.0 z) t (* y x)) (* a 2.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (fma((x / a), -0.5, ((((t * z) / a) / y) * 4.5)) * y) * -1.0;
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= 4e+278) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a * 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(fma(Float64(x / a), -0.5, Float64(Float64(Float64(Float64(t * z) / a) / y) * 4.5)) * y) * -1.0)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= 4e+278)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(N[(x / a), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * -1.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e+278], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 3.99999999999999985e278 < (*.f64 x y)

   1. Initial program 54.1%

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

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 97.0

          \[\leadsto \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1} \]

   if -inf.0 < (*.f64 x y) < 3.99999999999999985e278

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 4: 92.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{t \cdot z}{a}\\ t_2 := \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{t\_1}{y} \cdot 4.5\right) \cdot y\right) \cdot -1\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot x}{a}, 0.5, t\_1 \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (/ (* t z) a))
        (t_2 (* (* (fma (/ x a) -0.5 (* (/ t_1 y) 4.5)) y) -1.0)))
   (if (<= (* x y) (- INFINITY))
     t_2
     (if (<= (* x y) 2e+211) (fma (/ (* y x) a) 0.5 (* t_1 -4.5)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * z) / a;
	double t_2 = (fma((x / a), -0.5, ((t_1 / y) * 4.5)) * y) * -1.0;
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_2;
	} else if ((x * y) <= 2e+211) {
		tmp = fma(((y * x) / a), 0.5, (t_1 * -4.5));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(t * z) / a)
	t_2 = Float64(Float64(fma(Float64(x / a), -0.5, Float64(Float64(t_1 / y) * 4.5)) * y) * -1.0)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_2;
	elseif (Float64(x * y) <= 2e+211)
		tmp = fma(Float64(Float64(y * x) / a), 0.5, Float64(t_1 * -4.5));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(t * z), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x / a), $MachinePrecision] * -0.5 + N[(N[(t$95$1 / y), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * -1.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 2e+211], N[(N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision] * 0.5 + N[(t$95$1 * -4.5), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 1.9999999999999999e211 < (*.f64 x y)

   1. Initial program 58.8%

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

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 97.3

          \[\leadsto \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1} \]

   if -inf.0 < (*.f64 x y) < 1.9999999999999999e211

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 5: 87.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y \cdot x}{a}}{t}, 0.5, \frac{z}{a} \cdot -4.5\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (* (* (fma (/ x a) -0.5 (* (/ (/ (* t z) a) y) 4.5)) y) -1.0)))
   (if (<= (* x y) -5e+22)
     t_1
     (if (<= (* x y) 1e+96)
       (* (fma (/ (/ (* y x) a) t) 0.5 (* (/ z a) -4.5)) t)
       t_1))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (fma((x / a), -0.5, ((((t * z) / a) / y) * 4.5)) * y) * -1.0;
	double tmp;
	if ((x * y) <= -5e+22) {
		tmp = t_1;
	} else if ((x * y) <= 1e+96) {
		tmp = fma((((y * x) / a) / t), 0.5, ((z / a) * -4.5)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(fma(Float64(x / a), -0.5, Float64(Float64(Float64(Float64(t * z) / a) / y) * 4.5)) * y) * -1.0)
	tmp = 0.0
	if (Float64(x * y) <= -5e+22)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+96)
		tmp = Float64(fma(Float64(Float64(Float64(y * x) / a) / t), 0.5, Float64(Float64(z / a) * -4.5)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(N[(x / a), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * -1.0), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+22], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+96], N[(N[(N[(N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision] / t), $MachinePrecision] * 0.5 + N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999996e22 or 1.00000000000000005e96 < (*.f64 x y)

   1. Initial program 80.5%

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

   3. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.6

          \[\leadsto \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1} \]

   if -4.9999999999999996e22 < (*.f64 x y) < 1.00000000000000005e96

   1. Initial program 93.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

Alternative 6: 82.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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
 (* (* (fma (/ x a) -0.5 (* (/ (/ (* t z) a) y) 4.5)) y) -1.0))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (fma((x / a), -0.5, ((((t * z) / a) / y) * 4.5)) * y) * -1.0;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(fma(Float64(x / a), -0.5, Float64(Float64(Float64(Float64(t * z) / a) / y) * 4.5)) * y) * -1.0)
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(N[(N[(x / a), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * -1.0), $MachinePrecision]

Derivation

1. Initial program 88.8%

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

3. Taylor expanded in y around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 84.5

       \[\leadsto \left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1 \]

5. Applied rewrites 84.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{x}{a}, -0.5, \frac{\frac{t \cdot z}{a}}{y} \cdot 4.5\right) \cdot y\right) \cdot -1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))