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

Percentage Accurate: 79.3% → 91.4%

Time: 43.0s

Alternatives: 15

Speedup: 0.9×

• 36.4% 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: 79.3% 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: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{1}{c} \cdot \left(\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z} - t \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (/ 1.0 c) (- (/ (fma 9.0 (* x y) b) z) (* t (* 4.0 a))))))
   (if (<= z -1.5e+35)
     t_1
     (if (<= z 2.2e-23)
       (/ (fma (* z (* a -4.0)) t (fma y (* 9.0 x) b)) (* z c))
       t_1))))

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 = (1.0 / c) * ((fma(9.0, (x * y), b) / z) - (t * (4.0 * a)));
	double tmp;
	if (z <= -1.5e+35) {
		tmp = t_1;
	} else if (z <= 2.2e-23) {
		tmp = fma((z * (a * -4.0)), t, fma(y, (9.0 * x), 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(1.0 / c) * Float64(Float64(fma(9.0, Float64(x * y), b) / z) - Float64(t * Float64(4.0 * a))))
	tmp = 0.0
	if (z <= -1.5e+35)
		tmp = t_1;
	elseif (z <= 2.2e-23)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, fma(y, Float64(9.0 * x), 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(1.0 / c), $MachinePrecision] * N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] - N[(t * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e+35], t$95$1, If[LessEqual[z, 2.2e-23], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(y * N[(9.0 * x), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999995e35 or 2.1999999999999999e-23 < z

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 67.2

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

   4. Applied rewrites 67.2%

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

   6. Applied rewrites 82.5%

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

   7. Taylor expanded in z around inf

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \color{blue}{4} \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 92.5

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

   9. Applied rewrites 92.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-neg.f64 N/A

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

       10. clear-num N/A

           \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(c\right)}{\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)}}} \]

       11. associate-/r/ N/A

           \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(c\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(c\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)\right)} \]

   11. Applied rewrites 92.4%

       \[\leadsto \color{blue}{\frac{-1}{c} \cdot \left(t \cdot \left(4 \cdot a\right) - \frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}\right)} \]

   if -1.49999999999999995e35 < z < 2.1999999999999999e-23

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lift-*.f64 N/A

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

   6. Applied rewrites 97.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{c} \cdot \left(\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z} - t \cdot \left(4 \cdot a\right)\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z} - t \cdot \left(4 \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, 9 \cdot x, \mathsf{fma}\left(z, t \cdot \left(a \cdot -4\right), b\right)\right) \cdot \frac{1}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* 9.0 x)) (* a (* t (* z 4.0))))) (* z c))))
   (if (<= t_1 -2e+101)
     (/ (fma (* z (* a -4.0)) t (fma y (* 9.0 x) b)) (* z c))
     (if (<= t_1 0.0)
       (/ (/ (fma x (* 9.0 y) (fma a (* -4.0 (* z t)) b)) c) z)
       (if (<= t_1 INFINITY)
         (* (fma y (* 9.0 x) (fma z (* t (* a -4.0)) b)) (/ 1.0 (* z c)))
         (/ (fma -4.0 (* t a) (/ b z)) c))))))

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 * (9.0 * x)) - (a * (t * (z * 4.0))))) / (z * c);
	double tmp;
	if (t_1 <= -2e+101) {
		tmp = fma((z * (a * -4.0)), t, fma(y, (9.0 * x), b)) / (z * c);
	} else if (t_1 <= 0.0) {
		tmp = (fma(x, (9.0 * y), fma(a, (-4.0 * (z * t)), b)) / c) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (9.0 * x), fma(z, (t * (a * -4.0)), b)) * (1.0 / (z * c));
	} else {
		tmp = fma(-4.0, (t * a), (b / z)) / c;
	}
	return tmp;
}

Julia

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(9.0 * x)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -2e+101)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, fma(y, Float64(9.0 * x), b)) / Float64(z * c));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(a, Float64(-4.0 * Float64(z * t)), b)) / c) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y, Float64(9.0 * x), fma(z, Float64(t * Float64(a * -4.0)), b)) * Float64(1.0 / Float64(z * c)));
	else
		tmp = Float64(fma(-4.0, Float64(t * a), Float64(b / 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(9.0 * x), $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, -2e+101], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(y * N[(9.0 * x), $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] + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y * N[(9.0 * x), $MachinePrecision] + N[(z * N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / 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)) < -2e101

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 92.4

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

   4. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lift-*.f64 N/A

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

   6. Applied rewrites 88.5%

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

   if -2e101 < (/.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 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. 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 97.8%

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

   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 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 89.7

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot 9, \mathsf{fma}\left(z, t \cdot \left(-4 \cdot a\right), b\right)\right) \cdot \frac{1}{z \cdot c}} \]

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 1.6

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

   4. Applied rewrites 1.6%

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

   6. Applied rewrites 33.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 64.2

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

   9. Applied rewrites 64.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, 9 \cdot x, \mathsf{fma}\left(z, t \cdot \left(a \cdot -4\right), b\right)\right) \cdot \frac{1}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, 9 \cdot x, \mathsf{fma}\left(z, t \cdot \left(a \cdot -4\right), b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* 9.0 x)) (* a (* t (* z 4.0))))) (* z c))))
   (if (<= t_1 -1e+44)
     (/ (fma (* z (* a -4.0)) t (fma y (* 9.0 x) b)) (* z c))
     (if (<= t_1 INFINITY)
       (/ (/ (fma y (* 9.0 x) (fma z (* t (* a -4.0)) b)) z) c)
       (/ (fma -4.0 (* t a) (/ b z)) c)))))

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 * (9.0 * x)) - (a * (t * (z * 4.0))))) / (z * c);
	double tmp;
	if (t_1 <= -1e+44) {
		tmp = fma((z * (a * -4.0)), t, fma(y, (9.0 * x), b)) / (z * c);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (fma(y, (9.0 * x), fma(z, (t * (a * -4.0)), b)) / z) / c;
	} else {
		tmp = fma(-4.0, (t * a), (b / z)) / c;
	}
	return tmp;
}

Julia

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(9.0 * x)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e+44)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, fma(y, Float64(9.0 * x), b)) / Float64(z * c));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(fma(y, Float64(9.0 * x), fma(z, Float64(t * Float64(a * -4.0)), b)) / z) / c);
	else
		tmp = Float64(fma(-4.0, Float64(t * a), Float64(b / 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(9.0 * x), $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+44], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(y * N[(9.0 * x), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(y * N[(9.0 * x), $MachinePrecision] + N[(z * N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / 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)) < -1.0000000000000001e44

   1. Initial program 93.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lift-*.f64 N/A

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

   6. Applied rewrites 87.2%

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

   if -1.0000000000000001e44 < (/.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 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 83.7

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 89.9%

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

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 1.6

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

   4. Applied rewrites 1.6%

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

   6. Applied rewrites 33.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 64.2

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

   9. Applied rewrites 64.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, 9 \cdot x, \mathsf{fma}\left(z, t \cdot \left(a \cdot -4\right), b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ t_2 := y \cdot \left(9 \cdot x\right)\\ t_3 := \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

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 (* t a)) c))
        (t_2 (* y (* 9.0 x)))
        (t_3 (/ (* 9.0 (* x y)) (* z c))))
   (if (<= t_2 -5e+66)
     t_3
     (if (<= t_2 -2e-286)
       t_1
       (if (<= t_2 4e-320) (/ b (* z c)) (if (<= t_2 1e-21) t_1 t_3))))))

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 * (t * a)) / c;
	double t_2 = y * (9.0 * x);
	double t_3 = (9.0 * (x * y)) / (z * c);
	double tmp;
	if (t_2 <= -5e+66) {
		tmp = t_3;
	} else if (t_2 <= -2e-286) {
		tmp = t_1;
	} else if (t_2 <= 4e-320) {
		tmp = b / (z * c);
	} else if (t_2 <= 1e-21) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((-4.0d0) * (t * a)) / c
    t_2 = y * (9.0d0 * x)
    t_3 = (9.0d0 * (x * y)) / (z * c)
    if (t_2 <= (-5d+66)) then
        tmp = t_3
    else if (t_2 <= (-2d-286)) then
        tmp = t_1
    else if (t_2 <= 4d-320) then
        tmp = b / (z * c)
    else if (t_2 <= 1d-21) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

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 * (t * a)) / c;
	double t_2 = y * (9.0 * x);
	double t_3 = (9.0 * (x * y)) / (z * c);
	double tmp;
	if (t_2 <= -5e+66) {
		tmp = t_3;
	} else if (t_2 <= -2e-286) {
		tmp = t_1;
	} else if (t_2 <= 4e-320) {
		tmp = b / (z * c);
	} else if (t_2 <= 1e-21) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

[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 * (t * a)) / c
	t_2 = y * (9.0 * x)
	t_3 = (9.0 * (x * y)) / (z * c)
	tmp = 0
	if t_2 <= -5e+66:
		tmp = t_3
	elif t_2 <= -2e-286:
		tmp = t_1
	elif t_2 <= 4e-320:
		tmp = b / (z * c)
	elif t_2 <= 1e-21:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

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(t * a)) / c)
	t_2 = Float64(y * Float64(9.0 * x))
	t_3 = Float64(Float64(9.0 * Float64(x * y)) / Float64(z * c))
	tmp = 0.0
	if (t_2 <= -5e+66)
		tmp = t_3;
	elseif (t_2 <= -2e-286)
		tmp = t_1;
	elseif (t_2 <= 4e-320)
		tmp = Float64(b / Float64(z * c));
	elseif (t_2 <= 1e-21)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

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 * (t * a)) / c;
	t_2 = y * (9.0 * x);
	t_3 = (9.0 * (x * y)) / (z * c);
	tmp = 0.0;
	if (t_2 <= -5e+66)
		tmp = t_3;
	elseif (t_2 <= -2e-286)
		tmp = t_1;
	elseif (t_2 <= 4e-320)
		tmp = b / (z * c);
	elseif (t_2 <= 1e-21)
		tmp = t_1;
	else
		tmp = t_3;
	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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+66], t$95$3, If[LessEqual[t$95$2, -2e-286], t$95$1, If[LessEqual[t$95$2, 4e-320], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-21], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999991e66 or 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if -4.99999999999999991e66 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.0000000000000001e-286 or 3.99996e-320 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 73.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 56.1

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

   5. Applied rewrites 56.1%

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

   if -2.0000000000000001e-286 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99996e-320

   1. Initial program 92.6%

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

   3. Taylor expanded in 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 71.9

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

      1. distribute-rgt-neg-out N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{\color{blue}{0 - z \cdot c}}\right) \]

      5. lower--.f64 71.9

         \[\leadsto -\frac{b}{\color{blue}{0 - z \cdot c}} \]

   9. Applied rewrites 71.9%

      \[\leadsto -\frac{b}{\color{blue}{0 - z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{+66}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{-21}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ t_2 := y \cdot \left(9 \cdot x\right)\\ t_3 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+66}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

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 (* t a)) c))
        (t_2 (* y (* 9.0 x)))
        (t_3 (* 9.0 (/ (* x y) (* z c)))))
   (if (<= t_2 -5e+66)
     t_3
     (if (<= t_2 -2e-286)
       t_1
       (if (<= t_2 4e-320) (/ b (* z c)) (if (<= t_2 1e-21) t_1 t_3))))))

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 * (t * a)) / c;
	double t_2 = y * (9.0 * x);
	double t_3 = 9.0 * ((x * y) / (z * c));
	double tmp;
	if (t_2 <= -5e+66) {
		tmp = t_3;
	} else if (t_2 <= -2e-286) {
		tmp = t_1;
	} else if (t_2 <= 4e-320) {
		tmp = b / (z * c);
	} else if (t_2 <= 1e-21) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((-4.0d0) * (t * a)) / c
    t_2 = y * (9.0d0 * x)
    t_3 = 9.0d0 * ((x * y) / (z * c))
    if (t_2 <= (-5d+66)) then
        tmp = t_3
    else if (t_2 <= (-2d-286)) then
        tmp = t_1
    else if (t_2 <= 4d-320) then
        tmp = b / (z * c)
    else if (t_2 <= 1d-21) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

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 * (t * a)) / c;
	double t_2 = y * (9.0 * x);
	double t_3 = 9.0 * ((x * y) / (z * c));
	double tmp;
	if (t_2 <= -5e+66) {
		tmp = t_3;
	} else if (t_2 <= -2e-286) {
		tmp = t_1;
	} else if (t_2 <= 4e-320) {
		tmp = b / (z * c);
	} else if (t_2 <= 1e-21) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

[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 * (t * a)) / c
	t_2 = y * (9.0 * x)
	t_3 = 9.0 * ((x * y) / (z * c))
	tmp = 0
	if t_2 <= -5e+66:
		tmp = t_3
	elif t_2 <= -2e-286:
		tmp = t_1
	elif t_2 <= 4e-320:
		tmp = b / (z * c)
	elif t_2 <= 1e-21:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

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(t * a)) / c)
	t_2 = Float64(y * Float64(9.0 * x))
	t_3 = Float64(9.0 * Float64(Float64(x * y) / Float64(z * c)))
	tmp = 0.0
	if (t_2 <= -5e+66)
		tmp = t_3;
	elseif (t_2 <= -2e-286)
		tmp = t_1;
	elseif (t_2 <= 4e-320)
		tmp = Float64(b / Float64(z * c));
	elseif (t_2 <= 1e-21)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

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 * (t * a)) / c;
	t_2 = y * (9.0 * x);
	t_3 = 9.0 * ((x * y) / (z * c));
	tmp = 0.0;
	if (t_2 <= -5e+66)
		tmp = t_3;
	elseif (t_2 <= -2e-286)
		tmp = t_1;
	elseif (t_2 <= 4e-320)
		tmp = b / (z * c);
	elseif (t_2 <= 1e-21)
		tmp = t_1;
	else
		tmp = t_3;
	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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(9.0 * N[(N[(x * y), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+66], t$95$3, If[LessEqual[t$95$2, -2e-286], t$95$1, If[LessEqual[t$95$2, 4e-320], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-21], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999991e66 or 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 80.6

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

   4. Applied rewrites 80.6%

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if -4.99999999999999991e66 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.0000000000000001e-286 or 3.99996e-320 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 73.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 56.1

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

   5. Applied rewrites 56.1%

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

   if -2.0000000000000001e-286 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99996e-320

   1. Initial program 92.6%

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

   3. Taylor expanded in 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 71.9

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 71.9

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

   7. Applied rewrites 71.9%

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

      1. distribute-rgt-neg-out N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{\color{blue}{0 - z \cdot c}}\right) \]

      5. lower--.f64 71.9

         \[\leadsto -\frac{b}{\color{blue}{0 - z \cdot c}} \]

   9. Applied rewrites 71.9%

      \[\leadsto -\frac{b}{\color{blue}{0 - z \cdot c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{+66}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 4 \cdot 10^{-320}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{-21}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (if (<= t_1 -5000000000.0)
     (/ (/ (fma 9.0 (* x y) b) c) z)
     (if (<= t_1 1e+162)
       (/ (fma -4.0 (* t a) (/ b z)) c)
       (/ (fma a (* -4.0 (* z t)) (* 9.0 (* x y))) (* z c))))))

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 * (9.0 * x);
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = (fma(9.0, (x * y), b) / c) / z;
	} else if (t_1 <= 1e+162) {
		tmp = fma(-4.0, (t * a), (b / z)) / c;
	} else {
		tmp = fma(a, (-4.0 * (z * t)), (9.0 * (x * y))) / (z * c);
	}
	return tmp;
}

Julia

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(9.0 * x))
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / c) / z);
	elseif (t_1 <= 1e+162)
		tmp = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c);
	else
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), Float64(9.0 * Float64(x * y))) / Float64(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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+162], N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5e9

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. 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 78.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   if -5e9 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999994e161

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 82.6

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 89.0%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.7

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

   9. Applied rewrites 87.7%

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

   if 9.9999999999999994e161 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 84.4%

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

   3. Taylor expanded in b around 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 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (if (<= t_1 -5000000000.0)
     (/ (/ (fma 9.0 (* x y) b) c) z)
     (if (<= t_1 1e-21)
       (/ (fma -4.0 (* t a) (/ b z)) c)
       (/ (fma (* 9.0 x) y b) (* z c))))))

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 * (9.0 * x);
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = (fma(9.0, (x * y), b) / c) / z;
	} else if (t_1 <= 1e-21) {
		tmp = fma(-4.0, (t * a), (b / z)) / c;
	} else {
		tmp = fma((9.0 * x), y, b) / (z * c);
	}
	return tmp;
}

Julia

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(9.0 * x))
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / c) / z);
	elseif (t_1 <= 1e-21)
		tmp = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c);
	else
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5e9

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. 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 78.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   if -5e9 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 81.4

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 86.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 85.9

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

   4. Applied rewrites 85.9%

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in z around inf

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \color{blue}{4} \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.6

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

   9. Applied rewrites 82.6%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 73.6

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

   12. Applied rewrites 73.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (if (<= t_1 -5000000000.0)
     (/ (/ (fma 9.0 (* x y) b) c) z)
     (if (<= t_1 5e-39)
       (/ (fma a (* -4.0 (* z t)) b) (* z c))
       (/ (fma (* 9.0 x) y b) (* z c))))))

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 * (9.0 * x);
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = (fma(9.0, (x * y), b) / c) / z;
	} else if (t_1 <= 5e-39) {
		tmp = fma(a, (-4.0 * (z * t)), b) / (z * c);
	} else {
		tmp = fma((9.0 * x), y, b) / (z * c);
	}
	return tmp;
}

Julia

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(9.0 * x))
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / c) / z);
	elseif (t_1 <= 5e-39)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), b) / Float64(z * c));
	else
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e-39], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5e9

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. 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 78.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   if -5e9 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999998e-39

   1. Initial program 80.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 78.7

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

   5. Applied rewrites 78.7%

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

   if 4.9999999999999998e-39 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 86.5

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

   4. Applied rewrites 86.5%

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in z around inf

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \color{blue}{4} \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 83.4

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

   9. Applied rewrites 83.4%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 73.3

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

   12. Applied rewrites 73.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5000000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (fma -4.0 (* t a) (/ b z)) c)))
   (if (<= z -6e+140)
     t_1
     (if (<= z 2.05e+142)
       (/ (fma (* z (* a -4.0)) t (fma y (* 9.0 x) b)) (* z c))
       t_1))))

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(-4.0, (t * a), (b / z)) / c;
	double tmp;
	if (z <= -6e+140) {
		tmp = t_1;
	} else if (z <= 2.05e+142) {
		tmp = fma((z * (a * -4.0)), t, fma(y, (9.0 * x), 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c)
	tmp = 0.0
	if (z <= -6e+140)
		tmp = t_1;
	elseif (z <= 2.05e+142)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, fma(y, Float64(9.0 * x), 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -6e+140], t$95$1, If[LessEqual[z, 2.05e+142], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(y * N[(9.0 * x), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.99999999999999993e140 or 2.04999999999999991e142 < z

   1. Initial program 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 53.2

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

   4. Applied rewrites 53.2%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 77.8

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

   9. Applied rewrites 77.8%

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

   if -5.99999999999999993e140 < z < 2.04999999999999991e142

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 91.9

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

   4. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

      11. lift-*.f64 N/A

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

   6. Applied rewrites 91.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, \mathsf{fma}\left(y, 9 \cdot x, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (fma -4.0 (* t a) (/ b z)) c)))
   (if (<= z -1.45e+100)
     t_1
     (if (<= z 1.85e+135)
       (/ (fma (* 9.0 x) y (fma a (* -4.0 (* z t)) b)) (* z c))
       t_1))))

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(-4.0, (t * a), (b / z)) / c;
	double tmp;
	if (z <= -1.45e+100) {
		tmp = t_1;
	} else if (z <= 1.85e+135) {
		tmp = fma((9.0 * x), y, fma(a, (-4.0 * (z * t)), 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c)
	tmp = 0.0
	if (z <= -1.45e+100)
		tmp = t_1;
	elseif (z <= 1.85e+135)
		tmp = Float64(fma(Float64(9.0 * x), y, fma(a, Float64(-4.0 * Float64(z * t)), 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -1.45e+100], t$95$1, If[LessEqual[z, 1.85e+135], N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e100 or 1.84999999999999999e135 < z

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 57.1

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

   4. Applied rewrites 57.1%

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

   6. Applied rewrites 77.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 77.9

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

   9. Applied rewrites 77.9%

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

   if -1.45e100 < z < 1.84999999999999999e135

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l- N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 92.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* t a)) c)))
   (if (<= z -1.9e+87)
     t_1
     (if (<= z -1.2e-191)
       (/ (fma a (* -4.0 (* z t)) b) (* z c))
       (if (<= z 1.6e+116) (/ (fma (* 9.0 x) y b) (* z c)) t_1)))))

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 * (t * a)) / c;
	double tmp;
	if (z <= -1.9e+87) {
		tmp = t_1;
	} else if (z <= -1.2e-191) {
		tmp = fma(a, (-4.0 * (z * t)), b) / (z * c);
	} else if (z <= 1.6e+116) {
		tmp = fma((9.0 * x), 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(t * a)) / c)
	tmp = 0.0
	if (z <= -1.9e+87)
		tmp = t_1;
	elseif (z <= -1.2e-191)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), b) / Float64(z * c));
	elseif (z <= 1.6e+116)
		tmp = Float64(fma(Float64(9.0 * x), 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -1.9e+87], t$95$1, If[LessEqual[z, -1.2e-191], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+116], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.90000000000000006e87 or 1.6e116 < z

   1. Initial program 55.3%

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

   3. Taylor expanded in z around 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 64.0

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

   5. Applied rewrites 64.0%

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

   if -1.90000000000000006e87 < z < -1.2e-191

   1. Initial program 92.9%

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

   3. Taylor expanded in 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 79.0

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

   5. Applied rewrites 79.0%

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

   if -1.2e-191 < z < 1.6e116

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 91.8

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

   4. Applied rewrites 91.8%

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in z around inf

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \color{blue}{4} \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 87.3

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

   9. Applied rewrites 87.3%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 81.3

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

   12. Applied rewrites 81.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* t a)) c)))
   (if (<= z -3.1e+96)
     t_1
     (if (<= z 1.6e+116) (/ (fma (* 9.0 x) y b) (* z c)) t_1))))

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 * (t * a)) / c;
	double tmp;
	if (z <= -3.1e+96) {
		tmp = t_1;
	} else if (z <= 1.6e+116) {
		tmp = fma((9.0 * x), 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(t * a)) / c)
	tmp = 0.0
	if (z <= -3.1e+96)
		tmp = t_1;
	elseif (z <= 1.6e+116)
		tmp = Float64(fma(Float64(9.0 * x), 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -3.1e+96], t$95$1, If[LessEqual[z, 1.6e+116], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0999999999999998e96 or 1.6e116 < z

   1. Initial program 55.4%

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

   3. Taylor expanded in 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 64.3

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

   5. Applied rewrites 64.3%

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

   if -3.0999999999999998e96 < z < 1.6e116

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 92.2

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

   4. Applied rewrites 92.2%

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

   6. Applied rewrites 88.4%

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

   7. Taylor expanded in z around inf

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

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

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + \color{blue}{4} \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right)} + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right)} + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}\right)\right) + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{b + \color{blue}{9 \cdot \left(x \cdot y\right)}}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{b + 9 \cdot \color{blue}{\left(x \cdot y\right)}}{z}\right)\right) + 4 \cdot \left(a \cdot t\right)}{\mathsf{neg}\left(c\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)\right) + \color{blue}{4 \cdot \left(a \cdot t\right)}}{\mathsf{neg}\left(c\right)} \]

      11. lower-*.f64 87.7

          \[\leadsto \frac{\left(-\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) + 4 \cdot \color{blue}{\left(a \cdot t\right)}}{-c} \]

   9. Applied rewrites 87.7%

      \[\leadsto \frac{\color{blue}{\left(-\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) + 4 \cdot \left(a \cdot t\right)}}{-c} \]

   10. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{c \cdot z} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot y} + b}{c \cdot z} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9 \cdot x, y, b\right)}}{c \cdot z} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{9 \cdot x}, y, b\right)}{c \cdot z} \]

       6. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{\color{blue}{z \cdot c}} \]

       7. lower-*.f64 76.4

          \[\leadsto \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{\color{blue}{z \cdot c}} \]

   12. Applied rewrites 76.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* t a)) c)))
   (if (<= z -3.1e+96)
     t_1
     (if (<= z 1.6e+116) (/ (fma 9.0 (* x y) b) (* z c)) t_1))))

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 * (t * a)) / c;
	double tmp;
	if (z <= -3.1e+96) {
		tmp = t_1;
	} else if (z <= 1.6e+116) {
		tmp = fma(9.0, (x * 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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(t * a)) / c)
	tmp = 0.0
	if (z <= -3.1e+96)
		tmp = t_1;
	elseif (z <= 1.6e+116)
		tmp = Float64(fma(9.0, Float64(x * 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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -3.1e+96], t$95$1, If[LessEqual[z, 1.6e+116], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0999999999999998e96 or 1.6e116 < z

   1. Initial program 55.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 64.3

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -3.0999999999999998e96 < z < 1.6e116

   1. Initial program 92.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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 76.4

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* -4.0 (* t a)) c)))
   (if (<= a -3.4e-62) t_1 (if (<= a 8.2e+51) (/ b (* z c)) t_1))))

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 * (t * a)) / c;
	double tmp;
	if (a <= -3.4e-62) {
		tmp = t_1;
	} else if (a <= 8.2e+51) {
		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.
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) * (t * a)) / c
    if (a <= (-3.4d-62)) then
        tmp = t_1
    else if (a <= 8.2d+51) 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;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-4.0 * (t * a)) / c;
	double tmp;
	if (a <= -3.4e-62) {
		tmp = t_1;
	} else if (a <= 8.2e+51) {
		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])
def code(x, y, z, t, a, b, c):
	t_1 = (-4.0 * (t * a)) / c
	tmp = 0
	if a <= -3.4e-62:
		tmp = t_1
	elif a <= 8.2e+51:
		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])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-4.0 * Float64(t * a)) / c)
	tmp = 0.0
	if (a <= -3.4e-62)
		tmp = t_1;
	elseif (a <= 8.2e+51)
		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])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (-4.0 * (t * a)) / c;
	tmp = 0.0;
	if (a <= -3.4e-62)
		tmp = t_1;
	elseif (a <= 8.2e+51)
		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.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[a, -3.4e-62], t$95$1, If[LessEqual[a, 8.2e+51], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.39999999999999988e-62 or 8.20000000000000021e51 < a

   1. Initial program 78.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 59.4

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -3.39999999999999988e-62 < a < 8.20000000000000021e51

   1. Initial program 80.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 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 51.6

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \frac{b}{z \cdot c} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(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);
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.
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;
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])
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])
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])){:}
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.
code[x_, y_, z_, t_, a_, b_, c_] := N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Add Preprocessing

3. 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 36.7

      \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

5. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
6. Add Preprocessing

Developer Target 1: 80.8% 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 2024220 
(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)))