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

Percentage Accurate: 79.0% → 90.0%

Time: 16.9s

Alternatives: 20

Speedup: 0.7×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}{c}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 550:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* y (fma 9.0 (/ x z) (/ (fma -4.0 (* a t) (/ b z)) y))) c)))
   (if (<= z -8.5e+43)
     t_1
     (if (<= z 550.0)
       (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* z c))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (y * fma(9.0, (x / z), (fma(-4.0, (a * t), (b / z)) / y))) / c;
	double tmp;
	if (z <= -8.5e+43) {
		tmp = t_1;
	} else if (z <= 550.0) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(y * fma(9.0, Float64(x / z), Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / y))) / c)
	tmp = 0.0
	if (z <= -8.5e+43)
		tmp = t_1;
	elseif (z <= 550.0)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(y * N[(9.0 * N[(x / z), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -8.5e+43], t$95$1, If[LessEqual[z, 550.0], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e43 or 550 < z

   1. Initial program 55.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 92.9%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   if -8.5e43 < z < 550

   1. Initial program 97.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \color{blue}{\left(a \cdot t\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a\right) \cdot t} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a}, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot \color{blue}{-4}\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      20. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

   4. Applied rewrites 95.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}{c}\\ \mathbf{elif}\;z \leq 550:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ t_2 := \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, t\_2\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, t\_2\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c)))
        (t_2 (fma a (* -4.0 (* z t)) b)))
   (if (<= t_1 -1e-202)
     (/ (fma (* z -4.0) (* a t) (fma x (* 9.0 y) b)) (* z c))
     (if (<= t_1 5e+95)
       (/ (/ (fma x (* 9.0 y) t_2) z) c)
       (if (<= t_1 INFINITY)
         (/ (fma (* x 9.0) y t_2) (* z c))
         (fma (/ (* x 9.0) c) (/ y z) (/ (* -4.0 (* a t)) c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double t_2 = fma(a, (-4.0 * (z * t)), b);
	double tmp;
	if (t_1 <= -1e-202) {
		tmp = fma((z * -4.0), (a * t), fma(x, (9.0 * y), b)) / (z * c);
	} else if (t_1 <= 5e+95) {
		tmp = (fma(x, (9.0 * y), t_2) / z) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, t_2) / (z * c);
	} else {
		tmp = fma(((x * 9.0) / c), (y / z), ((-4.0 * (a * t)) / c));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	t_2 = fma(a, Float64(-4.0 * Float64(z * t)), b)
	tmp = 0.0
	if (t_1 <= -1e-202)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * t), fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	elseif (t_1 <= 5e+95)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_2) / z) / c);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, t_2) / Float64(z * c));
	else
		tmp = fma(Float64(Float64(x * 9.0) / c), Float64(y / z), Float64(Float64(-4.0 * Float64(a * t)) / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-202], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+95], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$2), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$2), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] / c), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-202

   1. Initial program 84.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-4}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, \color{blue}{t \cdot a}, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}\right)}{z \cdot c} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, \color{blue}{9 \cdot y}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z \cdot c} \]

   if -1e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 5.00000000000000025e95

   1. Initial program 67.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   if 5.00000000000000025e95 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 89.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 89.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y}}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{c \cdot z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{y}{z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot 9}{c}}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      14. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}}\right)\right) \]

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, -\frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c}\right)} \]

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c}\right) \]

      4. lower-*.f64 95.4

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c}\right) \]

   7. Applied rewrites 95.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ t_2 := \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, t\_2\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, t\_2\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c)))
        (t_2 (fma a (* -4.0 (* z t)) b)))
   (if (<= t_1 -1e-202)
     (/ (fma (* z -4.0) (* a t) (fma x (* 9.0 y) b)) (* z c))
     (if (<= t_1 5e+95)
       (/ (/ (fma x (* 9.0 y) t_2) z) c)
       (if (<= t_1 INFINITY)
         (/ (fma (* x 9.0) y t_2) (* z c))
         (fma 9.0 (* y (/ x (* z c))) (/ (* -4.0 (* a t)) c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double t_2 = fma(a, (-4.0 * (z * t)), b);
	double tmp;
	if (t_1 <= -1e-202) {
		tmp = fma((z * -4.0), (a * t), fma(x, (9.0 * y), b)) / (z * c);
	} else if (t_1 <= 5e+95) {
		tmp = (fma(x, (9.0 * y), t_2) / z) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, t_2) / (z * c);
	} else {
		tmp = fma(9.0, (y * (x / (z * c))), ((-4.0 * (a * t)) / c));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	t_2 = fma(a, Float64(-4.0 * Float64(z * t)), b)
	tmp = 0.0
	if (t_1 <= -1e-202)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * t), fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	elseif (t_1 <= 5e+95)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_2) / z) / c);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, t_2) / Float64(z * c));
	else
		tmp = fma(9.0, Float64(y * Float64(x / Float64(z * c))), Float64(Float64(-4.0 * Float64(a * t)) / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-202], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+95], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$2), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$2), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(y * N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-202

   1. Initial program 84.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-4}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, \color{blue}{t \cdot a}, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}\right)}{z \cdot c} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, \color{blue}{9 \cdot y}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z \cdot c} \]

   if -1e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 5.00000000000000025e95

   1. Initial program 67.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   if 5.00000000000000025e95 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 89.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 89.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + -4 \cdot \frac{a \cdot t}{c}} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + -4 \cdot \frac{a \cdot t}{c}\right)} \]

      3. metadata-eval N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \color{blue}{\left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{c \cdot z} \cdot \color{blue}{\left(\mathsf{neg}\left(-9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{c \cdot z} \cdot -9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot x}}{c \cdot z} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot \frac{x}{c \cdot z}\right)} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(\frac{x}{c \cdot z} \cdot -9\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right) \cdot y}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-9 \cdot \left(\frac{x}{c \cdot z} \cdot y\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-9\right)\right) \cdot \left(\frac{x}{c \cdot z} \cdot y\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      15. metadata-eval N/A

          \[\leadsto \color{blue}{9} \cdot \left(\frac{x}{c \cdot z} \cdot y\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{x}{z \cdot c} \cdot y, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)} \]

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ t_2 := \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, t\_2\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, t\_2\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c)))
        (t_2 (fma a (* -4.0 (* z t)) b)))
   (if (<= t_1 -1e-202)
     (/ (fma (* z -4.0) (* a t) (fma x (* 9.0 y) b)) (* z c))
     (if (<= t_1 0.0)
       (/ (/ (fma x (* 9.0 y) t_2) c) z)
       (if (<= t_1 INFINITY)
         (/ (fma (* x 9.0) y t_2) (* z c))
         (fma 9.0 (* y (/ x (* z c))) (/ (* -4.0 (* a t)) c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double t_2 = fma(a, (-4.0 * (z * t)), b);
	double tmp;
	if (t_1 <= -1e-202) {
		tmp = fma((z * -4.0), (a * t), fma(x, (9.0 * y), b)) / (z * c);
	} else if (t_1 <= 0.0) {
		tmp = (fma(x, (9.0 * y), t_2) / c) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, t_2) / (z * c);
	} else {
		tmp = fma(9.0, (y * (x / (z * c))), ((-4.0 * (a * t)) / c));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	t_2 = fma(a, Float64(-4.0 * Float64(z * t)), b)
	tmp = 0.0
	if (t_1 <= -1e-202)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * t), fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_2) / c) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, t_2) / Float64(z * c));
	else
		tmp = fma(9.0, Float64(y * Float64(x / Float64(z * c))), Float64(Float64(-4.0 * Float64(a * t)) / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-202], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$2), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$2), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(y * N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-202

   1. Initial program 84.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-4}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, \color{blue}{t \cdot a}, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}\right)}{z \cdot c} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, \color{blue}{9 \cdot y}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z \cdot c} \]

   if -1e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 37.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{c}}{z}} \]

   if 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 91.5

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + -4 \cdot \frac{a \cdot t}{c}} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + -4 \cdot \frac{a \cdot t}{c}\right)} \]

      3. metadata-eval N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \color{blue}{\left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{c \cdot z} \cdot \color{blue}{\left(\mathsf{neg}\left(-9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{c \cdot z} \cdot -9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot x}}{c \cdot z} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot \frac{x}{c \cdot z}\right)} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(\frac{x}{c \cdot z} \cdot -9\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right) \cdot y}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-9 \cdot \left(\frac{x}{c \cdot z} \cdot y\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-9\right)\right) \cdot \left(\frac{x}{c \cdot z} \cdot y\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      15. metadata-eval N/A

          \[\leadsto \color{blue}{9} \cdot \left(\frac{x}{c \cdot z} \cdot y\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{x}{z \cdot c} \cdot y, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)} \]

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c))))
   (if (<= t_1 -1e-202)
     (/ (fma (* z -4.0) (* a t) (fma x (* 9.0 y) b)) (* z c))
     (if (<= t_1 0.0)
       (/ (/ (fma 9.0 (* x y) b) z) c)
       (if (<= t_1 INFINITY)
         (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c))
         (fma 9.0 (* y (/ x (* z c))) (/ (* -4.0 (* a t)) c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double tmp;
	if (t_1 <= -1e-202) {
		tmp = fma((z * -4.0), (a * t), fma(x, (9.0 * y), b)) / (z * c);
	} else if (t_1 <= 0.0) {
		tmp = (fma(9.0, (x * y), b) / z) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c);
	} else {
		tmp = fma(9.0, (y * (x / (z * c))), ((-4.0 * (a * t)) / c));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e-202)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * t), fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c));
	else
		tmp = fma(9.0, Float64(y * Float64(x / Float64(z * c))), Float64(Float64(-4.0 * Float64(a * t)) / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-202], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(y * N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-202

   1. Initial program 84.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-4}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, \color{blue}{t \cdot a}, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}\right)}{z \cdot c} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, \color{blue}{9 \cdot y}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z \cdot c} \]

   if -1e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 37.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}}{c} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

      3. lower-*.f64 91.2

         \[\leadsto \frac{\frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z}}{c} \]

   7. Applied rewrites 91.2%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

   if 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 91.5

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + -4 \cdot \frac{a \cdot t}{c}} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + -4 \cdot \frac{a \cdot t}{c}\right)} \]

      3. metadata-eval N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \color{blue}{\left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{c \cdot z} \cdot \color{blue}{\left(\mathsf{neg}\left(-9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{c \cdot z} \cdot -9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot x}}{c \cdot z} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot \frac{x}{c \cdot z}\right)} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(\frac{x}{c \cdot z} \cdot -9\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right) \cdot y}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-9 \cdot \left(\frac{x}{c \cdot z} \cdot y\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-9\right)\right) \cdot \left(\frac{x}{c \cdot z} \cdot y\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      15. metadata-eval N/A

          \[\leadsto \color{blue}{9} \cdot \left(\frac{x}{c \cdot z} \cdot y\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{x}{z \cdot c} \cdot y, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)} \]

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 56.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{-4 \cdot \left(a \cdot t\right)}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c))))
   (if (<= t_1 -1e-202)
     (/ (fma (* z -4.0) (* a t) (fma x (* 9.0 y) b)) (* z c))
     (if (<= t_1 0.0)
       (/ (/ (fma 9.0 (* x y) b) z) c)
       (if (<= t_1 INFINITY)
         (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c))
         (/ (* y (/ (* x 9.0) z)) c))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double tmp;
	if (t_1 <= -1e-202) {
		tmp = fma((z * -4.0), (a * t), fma(x, (9.0 * y), b)) / (z * c);
	} else if (t_1 <= 0.0) {
		tmp = (fma(9.0, (x * y), b) / z) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c);
	} else {
		tmp = (y * ((x * 9.0) / z)) / c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e-202)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * t), fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c));
	else
		tmp = Float64(Float64(y * Float64(Float64(x * 9.0) / z)) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-202], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-202

   1. Initial program 84.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right), t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-4}, t \cdot a, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, \color{blue}{t \cdot a}, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      18. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}\right)}{z \cdot c} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, \color{blue}{9 \cdot y}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -4, t \cdot a, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z \cdot c} \]

   if -1e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 37.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}}{c} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

      3. lower-*.f64 91.2

         \[\leadsto \frac{\frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z}}{c} \]

   7. Applied rewrites 91.2%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

   if 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 91.5

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 44.2%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{\color{blue}{z}}}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* z c)))
        (t_2 (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c))))
   (if (<= t_1 -1e-196)
     t_2
     (if (<= t_1 0.0)
       (/ (/ (fma 9.0 (* x y) b) z) c)
       (if (<= t_1 INFINITY) t_2 (/ (* y (/ (* x 9.0) z)) c))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (z * c);
	double t_2 = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c);
	double tmp;
	if (t_1 <= -1e-196) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma(9.0, (x * y), b) / z) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (y * ((x * 9.0) / z)) / c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	t_2 = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e-196)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(y * Float64(Float64(x * 9.0) / z)) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-196], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(y * N[(N[(x * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1e-196 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 88.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      8. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      9. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right)\right) + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{a \cdot \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right)} + b\right)}{z \cdot c} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(a, \mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right), b\right)}\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right), b\right)\right)}{z \cdot c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot t\right), b\right)\right)}{z \cdot c} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot t\right)}\right), b\right)\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, \color{blue}{-4} \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c} \]

      21. lower-*.f64 88.0

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \color{blue}{\left(z \cdot t\right)}, b\right)\right)}{z \cdot c} \]

   4. Applied rewrites 88.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}}{z \cdot c} \]

   if -1e-196 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 40.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}}{c} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

      3. lower-*.f64 87.4

         \[\leadsto \frac{\frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z}}{c} \]

   7. Applied rewrites 87.4%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

   if +inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 0.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 44.2%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{\color{blue}{z}}}{c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ t_2 := \frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))) (t_2 (/ (* y (/ (* x 9.0) z)) c)))
   (if (<= t_1 -2e+262)
     t_2
     (if (<= t_1 -2e+46)
       (/ (/ (fma 9.0 (* x y) b) z) c)
       (if (<= t_1 4e-66)
         (/ (fma (* z (* a t)) -4.0 b) (* z c))
         (if (<= t_1 5e+207)
           (/ (fma a (* -4.0 (* z t)) (* 9.0 (* x y))) (* z c))
           t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * (x * 9.0);
	double t_2 = (y * ((x * 9.0) / z)) / c;
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_2;
	} else if (t_1 <= -2e+46) {
		tmp = (fma(9.0, (x * y), b) / z) / c;
	} else if (t_1 <= 4e-66) {
		tmp = fma((z * (a * t)), -4.0, b) / (z * c);
	} else if (t_1 <= 5e+207) {
		tmp = fma(a, (-4.0 * (z * t)), (9.0 * (x * y))) / (z * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * Float64(x * 9.0))
	t_2 = Float64(Float64(y * Float64(Float64(x * 9.0) / z)) / c)
	tmp = 0.0
	if (t_1 <= -2e+262)
		tmp = t_2;
	elseif (t_1 <= -2e+46)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c);
	elseif (t_1 <= 4e-66)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(z * c));
	elseif (t_1 <= 5e+207)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), Float64(9.0 * Float64(x * y))) / Float64(z * c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(x * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+262], t$95$2, If[LessEqual[t$95$1, -2e+46], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 4e-66], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+207], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e262 or 4.9999999999999999e207 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 61.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 92.5%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.3%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{\color{blue}{z}}}{c} \]

   if -2e262 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e46

   1. Initial program 73.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}}{c} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

      3. lower-*.f64 73.7

         \[\leadsto \frac{\frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z}}{c} \]

   7. Applied rewrites 73.7%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

   if -2e46 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.9999999999999999e-66

   1. Initial program 81.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. lower-*.f64 78.9

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Applied rewrites 78.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, \color{blue}{-4}, b\right)}{z \cdot c} \]

   if 3.9999999999999999e-66 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999999e207

   1. Initial program 81.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

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

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{9 \cdot \left(x \cdot y\right) + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + 9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + 9 \cdot \left(x \cdot y\right)}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + 9 \cdot \left(x \cdot y\right)}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + 9 \cdot \left(x \cdot y\right)}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), 9 \cdot \left(x \cdot y\right)\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, \color{blue}{9 \cdot \left(x \cdot y\right)}\right)}{z \cdot c} \]

      12. lower-*.f64 72.9

          \[\leadsto \frac{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, 9 \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z \cdot c} \]

   5. Applied rewrites 72.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 4 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ t_2 := \frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))) (t_2 (/ (* y (/ (* x 9.0) z)) c)))
   (if (<= t_1 -2e+262)
     t_2
     (if (<= t_1 -2e+46)
       (/ (/ (fma 9.0 (* x y) b) z) c)
       (if (<= t_1 5e+78)
         (/ (fma (* z (* a t)) -4.0 b) (* z c))
         (if (<= t_1 5e+207) (/ (/ (* 9.0 (* x y)) c) z) t_2))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * (x * 9.0);
	double t_2 = (y * ((x * 9.0) / z)) / c;
	double tmp;
	if (t_1 <= -2e+262) {
		tmp = t_2;
	} else if (t_1 <= -2e+46) {
		tmp = (fma(9.0, (x * y), b) / z) / c;
	} else if (t_1 <= 5e+78) {
		tmp = fma((z * (a * t)), -4.0, b) / (z * c);
	} else if (t_1 <= 5e+207) {
		tmp = ((9.0 * (x * y)) / c) / z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * Float64(x * 9.0))
	t_2 = Float64(Float64(y * Float64(Float64(x * 9.0) / z)) / c)
	tmp = 0.0
	if (t_1 <= -2e+262)
		tmp = t_2;
	elseif (t_1 <= -2e+46)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c);
	elseif (t_1 <= 5e+78)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(z * c));
	elseif (t_1 <= 5e+207)
		tmp = Float64(Float64(Float64(9.0 * Float64(x * y)) / c) / z);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(x * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+262], t$95$2, If[LessEqual[t$95$1, -2e+46], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 5e+78], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+207], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e262 or 4.9999999999999999e207 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 61.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 92.5%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.3%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{\color{blue}{z}}}{c} \]

   if -2e262 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e46

   1. Initial program 73.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}}{c} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

      3. lower-*.f64 73.7

         \[\leadsto \frac{\frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z}}{c} \]

   7. Applied rewrites 73.7%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z}}{c} \]

   if -2e46 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999984e78

   1. Initial program 82.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. lower-*.f64 75.6

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Applied rewrites 75.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, \color{blue}{-4}, b\right)}{z \cdot c} \]

   if 4.99999999999999984e78 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999999e207

   1. Initial program 76.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y}}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{c \cdot z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{y}{z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot 9}{c}}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      14. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}}\right)\right) \]

   4. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, -\frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{\frac{x \cdot y}{c}}, \frac{b}{c}\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{\color{blue}{x \cdot y}}{c}, \frac{b}{c}\right)}{z} \]

      5. lower-/.f64 76.5

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \color{blue}{\frac{b}{c}}\right)}{z} \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}{z}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.5%

       \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ t_2 := \frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))) (t_2 (/ (* y (/ (* x 9.0) z)) c)))
   (if (<= t_1 -2e+46)
     t_2
     (if (<= t_1 5e+78)
       (/ (fma (* z (* a t)) -4.0 b) (* z c))
       (if (<= t_1 5e+207) (/ (/ (* 9.0 (* x y)) c) z) t_2)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * (x * 9.0);
	double t_2 = (y * ((x * 9.0) / z)) / c;
	double tmp;
	if (t_1 <= -2e+46) {
		tmp = t_2;
	} else if (t_1 <= 5e+78) {
		tmp = fma((z * (a * t)), -4.0, b) / (z * c);
	} else if (t_1 <= 5e+207) {
		tmp = ((9.0 * (x * y)) / c) / z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * Float64(x * 9.0))
	t_2 = Float64(Float64(y * Float64(Float64(x * 9.0) / z)) / c)
	tmp = 0.0
	if (t_1 <= -2e+46)
		tmp = t_2;
	elseif (t_1 <= 5e+78)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(z * c));
	elseif (t_1 <= 5e+207)
		tmp = Float64(Float64(Float64(9.0 * Float64(x * y)) / c) / z);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(x * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+46], t$95$2, If[LessEqual[t$95$1, 5e+78], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+207], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e46 or 4.9999999999999999e207 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 66.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(-4 \cdot \frac{a \cdot t}{y} + \left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right)\right)}}{c} \]

      2. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\left(9 \cdot \frac{x}{z} + \frac{b}{y \cdot z}\right) + -4 \cdot \frac{a \cdot t}{y}\right)}}{c} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + -4 \cdot \frac{a \cdot t}{y}\right)\right)}}{c} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{y}}\right)\right)}{c} \]

      5. metadata-eval N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

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

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \frac{\color{blue}{\mathsf{neg}\left(4 \cdot \left(a \cdot t\right)\right)}}{y}\right)\right)}{c} \]

      7. distribute-frac-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{y \cdot z} + \color{blue}{\left(\mathsf{neg}\left(\frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}\right)\right)}{c} \]

      8. sub-neg N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\left(\frac{b}{y \cdot z} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)}\right)}{c} \]

      9. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\frac{b}{\color{blue}{z \cdot y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      10. associate-/r* N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \left(\color{blue}{\frac{\frac{b}{z}}{y}} - \frac{4 \cdot \left(a \cdot t\right)}{y}\right)\right)}{c} \]

      11. div-sub N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \color{blue}{\frac{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}{y}}\right)}{c} \]

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

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{y}\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{y}\right)}{c} \]

      14. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(9 \cdot \frac{x}{z} + \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{y}\right)}{c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(9, \frac{x}{z}, \frac{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}{y}\right)}}{c} \]

   7. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(9, \frac{x}{z}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{y}\right)}}{c} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 73.5%

       \[\leadsto \frac{y \cdot \frac{9 \cdot x}{\color{blue}{z}}}{c} \]

   if -2e46 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999984e78

   1. Initial program 82.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. lower-*.f64 75.6

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Applied rewrites 75.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, \color{blue}{-4}, b\right)}{z \cdot c} \]

   if 4.99999999999999984e78 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999999e207

   1. Initial program 76.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y}}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{c \cdot z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{y}{z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot 9}{c}}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      14. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}}\right)\right) \]

   4. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, -\frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{\frac{x \cdot y}{c}}, \frac{b}{c}\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{\color{blue}{x \cdot y}}{c}, \frac{b}{c}\right)}{z} \]

      5. lower-/.f64 76.5

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \color{blue}{\frac{b}{c}}\right)}{z} \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}{z}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.5%

       \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x \cdot 9}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 9.0 (* y (/ x (* z c))) (/ (fma -4.0 (* a t) (/ b z)) c))))
   (if (<= z -4.6e+24)
     t_1
     (if (<= z 1.35e-77)
       (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* z c))
       t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(9.0, (y * (x / (z * c))), (fma(-4.0, (a * t), (b / z)) / c));
	double tmp;
	if (z <= -4.6e+24) {
		tmp = t_1;
	} else if (z <= 1.35e-77) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = fma(9.0, Float64(y * Float64(x / Float64(z * c))), Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c))
	tmp = 0.0
	if (z <= -4.6e+24)
		tmp = t_1;
	elseif (z <= 1.35e-77)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(9.0 * N[(y * N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e+24], t$95$1, If[LessEqual[z, 1.35e-77], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5999999999999998e24 or 1.35e-77 < z

   1. Initial program 59.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + -4 \cdot \frac{a \cdot t}{c}} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + -4 \cdot \frac{a \cdot t}{c}\right)} \]

      3. metadata-eval N/A

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \left(\frac{b}{c \cdot z} + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto 9 \cdot \frac{x \cdot y}{c \cdot z} + \color{blue}{\left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot y}{c \cdot z} \cdot \color{blue}{\left(\mathsf{neg}\left(-9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{c \cdot z} \cdot -9\right)\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot x}}{c \cdot z} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      9. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot \frac{x}{c \cdot z}\right)} \cdot -9\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(\frac{x}{c \cdot z} \cdot -9\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-9 \cdot \frac{x}{c \cdot z}\right) \cdot y}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-9 \cdot \left(\frac{x}{c \cdot z} \cdot y\right)}\right)\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-9\right)\right) \cdot \left(\frac{x}{c \cdot z} \cdot y\right)} + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

      15. metadata-eval N/A

          \[\leadsto \color{blue}{9} \cdot \left(\frac{x}{c \cdot z} \cdot y\right) + \left(\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}\right) \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{x}{z \cdot c} \cdot y, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)} \]

   if -4.5999999999999998e24 < z < 1.35e-77

   1. Initial program 98.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \color{blue}{\left(a \cdot t\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a\right) \cdot t} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a}, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot \color{blue}{-4}\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{\left(x \cdot 9\right) \cdot y} + b\right)}{z \cdot c} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{\left(x \cdot 9\right)} \cdot y + b\right)}{z \cdot c} \]

      20. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(z \cdot -4\right) \cdot a, t, \color{blue}{x \cdot \left(9 \cdot y\right)} + b\right)}{z \cdot c} \]

   4. Applied rewrites 96.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, y \cdot \frac{x}{z \cdot c}, \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ t_2 := \frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))) (t_2 (/ (/ (* 9.0 (* x y)) c) z)))
   (if (<= t_1 -2e+46)
     t_2
     (if (<= t_1 5e+78) (/ (fma (* z (* a t)) -4.0 b) (* z c)) t_2))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * (x * 9.0);
	double t_2 = ((9.0 * (x * y)) / c) / z;
	double tmp;
	if (t_1 <= -2e+46) {
		tmp = t_2;
	} else if (t_1 <= 5e+78) {
		tmp = fma((z * (a * t)), -4.0, b) / (z * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * Float64(x * 9.0))
	t_2 = Float64(Float64(Float64(9.0 * Float64(x * y)) / c) / z)
	tmp = 0.0
	if (t_1 <= -2e+46)
		tmp = t_2;
	elseif (t_1 <= 5e+78)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(z * c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+46], t$95$2, If[LessEqual[t$95$1, 5e+78], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2e46 or 4.99999999999999984e78 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 68.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y}}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{c \cdot z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{y}{z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot 9}{c}}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      14. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}}\right)\right) \]

   4. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, -\frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{\frac{x \cdot y}{c}}, \frac{b}{c}\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{\color{blue}{x \cdot y}}{c}, \frac{b}{c}\right)}{z} \]

      5. lower-/.f64 68.4

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \color{blue}{\frac{b}{c}}\right)}{z} \]

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}{z}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 66.9%

       \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z} \]

   if -2e46 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999984e78

   1. Initial program 82.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. lower-*.f64 75.6

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Applied rewrites 75.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))))
   (if (<= t_1 -1.0)
     (/ (fma 9.0 (* x y) b) (* z c))
     (if (<= t_1 5e+78)
       (/ (fma a (* -4.0 (* z t)) b) (* z c))
       (/ (/ (* 9.0 (* x y)) c) z)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * (x * 9.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = fma(9.0, (x * y), b) / (z * c);
	} else if (t_1 <= 5e+78) {
		tmp = fma(a, (-4.0 * (z * t)), b) / (z * c);
	} else {
		tmp = ((9.0 * (x * y)) / c) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c));
	elseif (t_1 <= 5e+78)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), b) / Float64(z * c));
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(x * y)) / c) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+78], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1

   1. Initial program 64.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z \cdot c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. lower-*.f64 57.0

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Applied rewrites 57.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

   if -1 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999984e78

   1. Initial program 82.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. lower-*.f64 76.5

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Applied rewrites 76.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]

   if 4.99999999999999984e78 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 9\right) \cdot y}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y}}{z \cdot c} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{c \cdot z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{y}{z}} + \left(\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot 9}{c}}, \frac{y}{z}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \color{blue}{\frac{y}{z}}, \mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)\right) \]

      14. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \color{blue}{\mathsf{neg}\left(\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}\right)}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, \mathsf{neg}\left(\color{blue}{\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b}{z \cdot c}}\right)\right) \]

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot 9}{c}, \frac{y}{z}, -\frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{\frac{x \cdot y}{c}}, \frac{b}{c}\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{\color{blue}{x \cdot y}}{c}, \frac{b}{c}\right)}{z} \]

      5. lower-/.f64 76.2

         \[\leadsto \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \color{blue}{\frac{b}{c}}\right)}{z} \]

   7. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}{z}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 76.6%

       \[\leadsto \frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot \left(x \cdot y\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{t\_1}{c}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{9}{z \cdot c}\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -4.0 (* a t))))
   (if (<= z -4.8e+78)
     (/ t_1 c)
     (if (<= z 4.2e-277)
       (* x (* y (/ 9.0 (* z c))))
       (if (<= z 8.8e-8) (/ b (* z c)) (* t_1 (/ 1.0 c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if (z <= -4.8e+78) {
		tmp = t_1 / c;
	} else if (z <= 4.2e-277) {
		tmp = x * (y * (9.0 / (z * c)));
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1 * (1.0 / c);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * (a * t)
    if (z <= (-4.8d+78)) then
        tmp = t_1 / c
    else if (z <= 4.2d-277) then
        tmp = x * (y * (9.0d0 / (z * c)))
    else if (z <= 8.8d-8) then
        tmp = b / (z * c)
    else
        tmp = t_1 * (1.0d0 / c)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if (z <= -4.8e+78) {
		tmp = t_1 / c;
	} else if (z <= 4.2e-277) {
		tmp = x * (y * (9.0 / (z * c)));
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1 * (1.0 / c);
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = -4.0 * (a * t)
	tmp = 0
	if z <= -4.8e+78:
		tmp = t_1 / c
	elif z <= 4.2e-277:
		tmp = x * (y * (9.0 / (z * c)))
	elif z <= 8.8e-8:
		tmp = b / (z * c)
	else:
		tmp = t_1 * (1.0 / c)
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(-4.0 * Float64(a * t))
	tmp = 0.0
	if (z <= -4.8e+78)
		tmp = Float64(t_1 / c);
	elseif (z <= 4.2e-277)
		tmp = Float64(x * Float64(y * Float64(9.0 / Float64(z * c))));
	elseif (z <= 8.8e-8)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = Float64(t_1 * Float64(1.0 / c));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = -4.0 * (a * t);
	tmp = 0.0;
	if (z <= -4.8e+78)
		tmp = t_1 / c;
	elseif (z <= 4.2e-277)
		tmp = x * (y * (9.0 / (z * c)));
	elseif (z <= 8.8e-8)
		tmp = b / (z * c);
	else
		tmp = t_1 * (1.0 / c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.8e+78], N[(t$95$1 / c), $MachinePrecision], If[LessEqual[z, 4.2e-277], N[(x * N[(y * N[(9.0 / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.8e-8], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.7999999999999997e78

   1. Initial program 48.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 59.3

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -4.7999999999999997e78 < z < 4.1999999999999999e-277

   1. Initial program 97.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{c \cdot z}\right)} \cdot 9 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{c \cdot z} \cdot 9\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{9 \cdot y}{c \cdot z}} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot 9}}{c \cdot z} \]

      8. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{9}{c \cdot z}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{9}{c \cdot z}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{9}{c \cdot z}}\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \frac{9}{\color{blue}{z \cdot c}}\right) \]

      12. lower-*.f64 55.6

          \[\leadsto x \cdot \left(y \cdot \frac{9}{\color{blue}{z \cdot c}}\right) \]

   7. Applied rewrites 55.6%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{9}{z \cdot c}\right)} \]

   if 4.1999999999999999e-277 < z < 8.7999999999999994e-8

   1. Initial program 93.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]

   if 8.7999999999999994e-8 < z

   1. Initial program 58.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z} \cdot \frac{1}{c}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z} \cdot \frac{1}{c}} \]

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z} \cdot \frac{1}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]

      2. lower-*.f64 54.2

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot t\right)}\right) \cdot \frac{1}{c} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{9}{z \cdot c}\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* a t)) c)))
   (if (<= z -4.8e+78)
     t_1
     (if (<= z 4.2e-277)
       (* x (* y (/ 9.0 (* z c))))
       (if (<= z 8.8e-8) (/ b (* z c)) t_1)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-4.0 * (a * t)) / c;
	double tmp;
	if (z <= -4.8e+78) {
		tmp = t_1;
	} else if (z <= 4.2e-277) {
		tmp = x * (y * (9.0 / (z * c)));
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-4.0d0) * (a * t)) / c
    if (z <= (-4.8d+78)) then
        tmp = t_1
    else if (z <= 4.2d-277) then
        tmp = x * (y * (9.0d0 / (z * c)))
    else if (z <= 8.8d-8) then
        tmp = b / (z * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-4.0 * (a * t)) / c;
	double tmp;
	if (z <= -4.8e+78) {
		tmp = t_1;
	} else if (z <= 4.2e-277) {
		tmp = x * (y * (9.0 / (z * c)));
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (-4.0 * (a * t)) / c
	tmp = 0
	if z <= -4.8e+78:
		tmp = t_1
	elif z <= 4.2e-277:
		tmp = x * (y * (9.0 / (z * c)))
	elif z <= 8.8e-8:
		tmp = b / (z * c)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(a * t)) / c)
	tmp = 0.0
	if (z <= -4.8e+78)
		tmp = t_1;
	elseif (z <= 4.2e-277)
		tmp = Float64(x * Float64(y * Float64(9.0 / Float64(z * c))));
	elseif (z <= 8.8e-8)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (-4.0 * (a * t)) / c;
	tmp = 0.0;
	if (z <= -4.8e+78)
		tmp = t_1;
	elseif (z <= 4.2e-277)
		tmp = x * (y * (9.0 / (z * c)));
	elseif (z <= 8.8e-8)
		tmp = b / (z * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -4.8e+78], t$95$1, If[LessEqual[z, 4.2e-277], N[(x * N[(y * N[(9.0 / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.8e-8], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.7999999999999997e78 or 8.7999999999999994e-8 < z

   1. Initial program 53.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 56.3

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -4.7999999999999997e78 < z < 4.1999999999999999e-277

   1. Initial program 97.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{c \cdot z}\right)} \cdot 9 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{c \cdot z} \cdot 9\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{9 \cdot y}{c \cdot z}} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot 9}}{c \cdot z} \]

      8. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{9}{c \cdot z}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{9}{c \cdot z}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{9}{c \cdot z}}\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \left(y \cdot \frac{9}{\color{blue}{z \cdot c}}\right) \]

      12. lower-*.f64 55.6

          \[\leadsto x \cdot \left(y \cdot \frac{9}{\color{blue}{z \cdot c}}\right) \]

   7. Applied rewrites 55.6%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{9}{z \cdot c}\right)} \]

   if 4.1999999999999999e-277 < z < 8.7999999999999994e-8

   1. Initial program 93.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+128}:\\ \;\;\;\;\frac{t\_1}{c}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -4.0 (* a t))))
   (if (<= z -2.1e+128)
     (/ t_1 c)
     (if (<= z 2.3e+117) (/ (fma 9.0 (* x y) b) (* z c)) (* t_1 (/ 1.0 c))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if (z <= -2.1e+128) {
		tmp = t_1 / c;
	} else if (z <= 2.3e+117) {
		tmp = fma(9.0, (x * y), b) / (z * c);
	} else {
		tmp = t_1 * (1.0 / c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(-4.0 * Float64(a * t))
	tmp = 0.0
	if (z <= -2.1e+128)
		tmp = Float64(t_1 / c);
	elseif (z <= 2.3e+117)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c));
	else
		tmp = Float64(t_1 * Float64(1.0 / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+128], N[(t$95$1 / c), $MachinePrecision], If[LessEqual[z, 2.3e+117], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e128

   1. Initial program 35.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 65.4

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -2.1e128 < z < 2.29999999999999988e117

   1. Initial program 92.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z \cdot c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. lower-*.f64 74.1

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Applied rewrites 74.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

   if 2.29999999999999988e117 < z

   1. Initial program 50.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z} \cdot \frac{1}{c}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z} \cdot \frac{1}{c}} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z} \cdot \frac{1}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]

      2. lower-*.f64 61.1

         \[\leadsto \left(-4 \cdot \color{blue}{\left(a \cdot t\right)}\right) \cdot \frac{1}{c} \]

   7. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot t\right)\right)} \cdot \frac{1}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 47.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;b \leq 950000000000:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= b -1.5e+48)
   (/ b (* z c))
   (if (<= b 950000000000.0) (/ (* -4.0 (* a t)) c) (/ (/ b z) c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+48) {
		tmp = b / (z * c);
	} else if (b <= 950000000000.0) {
		tmp = (-4.0 * (a * t)) / c;
	} else {
		tmp = (b / z) / c;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d+48)) then
        tmp = b / (z * c)
    else if (b <= 950000000000.0d0) then
        tmp = ((-4.0d0) * (a * t)) / c
    else
        tmp = (b / z) / c
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+48) {
		tmp = b / (z * c);
	} else if (b <= 950000000000.0) {
		tmp = (-4.0 * (a * t)) / c;
	} else {
		tmp = (b / z) / c;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if b <= -1.5e+48:
		tmp = b / (z * c)
	elif b <= 950000000000.0:
		tmp = (-4.0 * (a * t)) / c
	else:
		tmp = (b / z) / c
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (b <= -1.5e+48)
		tmp = Float64(b / Float64(z * c));
	elseif (b <= 950000000000.0)
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c);
	else
		tmp = Float64(Float64(b / z) / c);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+48)
		tmp = b / (z * c);
	elseif (b <= 950000000000.0)
		tmp = (-4.0 * (a * t)) / c;
	else
		tmp = (b / z) / c;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[b, -1.5e+48], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 950000000000.0], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b / z), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e48

   1. Initial program 79.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 64.9

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]

   if -1.5e48 < b < 9.5e11

   1. Initial program 72.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 49.7

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if 9.5e11 < b

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]
   6. Step-by-step derivation

      1. lower-/.f64 58.4

         \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]

   7. Applied rewrites 58.4%

      \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;b \leq 950000000000:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= b -1.5e+48)
   (/ b (* z c))
   (if (<= b 950000000000.0) (/ (* -4.0 (* a t)) c) (/ (/ b c) z))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+48) {
		tmp = b / (z * c);
	} else if (b <= 950000000000.0) {
		tmp = (-4.0 * (a * t)) / c;
	} else {
		tmp = (b / c) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d+48)) then
        tmp = b / (z * c)
    else if (b <= 950000000000.0d0) then
        tmp = ((-4.0d0) * (a * t)) / c
    else
        tmp = (b / c) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+48) {
		tmp = b / (z * c);
	} else if (b <= 950000000000.0) {
		tmp = (-4.0 * (a * t)) / c;
	} else {
		tmp = (b / c) / z;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if b <= -1.5e+48:
		tmp = b / (z * c)
	elif b <= 950000000000.0:
		tmp = (-4.0 * (a * t)) / c
	else:
		tmp = (b / c) / z
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (b <= -1.5e+48)
		tmp = Float64(b / Float64(z * c));
	elseif (b <= 950000000000.0)
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c);
	else
		tmp = Float64(Float64(b / c) / z);
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+48)
		tmp = b / (z * c);
	elseif (b <= 950000000000.0)
		tmp = (-4.0 * (a * t)) / c;
	else
		tmp = (b / c) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[b, -1.5e+48], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 950000000000.0], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b / c), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e48

   1. Initial program 79.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 64.9

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]

   if -1.5e48 < b < 9.5e11

   1. Initial program 72.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 49.7

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if 9.5e11 < b

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 54.4

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

      \[\leadsto \frac{\frac{b}{c}}{\color{blue}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 49.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* a t)) c)))
   (if (<= z -3.2e+61) t_1 (if (<= z 8.8e-8) (/ b (* z c)) t_1))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-4.0 * (a * t)) / c;
	double tmp;
	if (z <= -3.2e+61) {
		tmp = t_1;
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-4.0d0) * (a * t)) / c
    if (z <= (-3.2d+61)) then
        tmp = t_1
    else if (z <= 8.8d-8) then
        tmp = b / (z * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-4.0 * (a * t)) / c;
	double tmp;
	if (z <= -3.2e+61) {
		tmp = t_1;
	} else if (z <= 8.8e-8) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (-4.0 * (a * t)) / c
	tmp = 0
	if z <= -3.2e+61:
		tmp = t_1
	elif z <= 8.8e-8:
		tmp = b / (z * c)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(a * t)) / c)
	tmp = 0.0
	if (z <= -3.2e+61)
		tmp = t_1;
	elseif (z <= 8.8e-8)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (-4.0 * (a * t)) / c;
	tmp = 0.0;
	if (z <= -3.2e+61)
		tmp = t_1;
	elseif (z <= 8.8e-8)
		tmp = b / (z * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -3.2e+61], t$95$1, If[LessEqual[z, 8.8e-8], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999998e61 or 8.7999999999999994e-8 < z

   1. Initial program 55.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 53.7

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -3.1999999999999998e61 < z < 8.7999999999999994e-8

   1. Initial program 97.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 53.7

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 34.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    code = b / (z * c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   3. lower-*.f64 35.9

      \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

5. Applied rewrites 35.9%

   \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
6. Add Preprocessing

Developer Target 1: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    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), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -220031348160821/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 365902434742109/31250000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 28768236795461370000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 138385150424563190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c)))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))