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

Percentage Accurate: 80.3% → 91.1%

Time: 13.8s

Alternatives: 19

Speedup: 0.8×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.3% 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: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \mathsf{fma}\left(\frac{y \cdot x}{z}, 9, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z \cdot c} \cdot 9, y, \mathsf{fma}\left(\frac{a}{c} \cdot t, -4, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= z -2.4e-67)
   (/ (fma (* a t) -4.0 (fma (/ (* y x) z) 9.0 (/ b z))) c)
   (if (<= z 1.75e-20)
     (/ (fma (* y 9.0) x (fma (* (* -4.0 z) a) t b)) (* z c))
     (fma (* (/ x (* z c)) 9.0) y (fma (* (/ a c) t) -4.0 (/ 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 tmp;
	if (z <= -2.4e-67) {
		tmp = fma((a * t), -4.0, fma(((y * x) / z), 9.0, (b / z))) / c;
	} else if (z <= 1.75e-20) {
		tmp = fma((y * 9.0), x, fma(((-4.0 * z) * a), t, b)) / (z * c);
	} else {
		tmp = fma(((x / (z * c)) * 9.0), y, fma(((a / c) * t), -4.0, (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)
	tmp = 0.0
	if (z <= -2.4e-67)
		tmp = Float64(fma(Float64(a * t), -4.0, fma(Float64(Float64(y * x) / z), 9.0, Float64(b / z))) / c);
	elseif (z <= 1.75e-20)
		tmp = Float64(fma(Float64(y * 9.0), x, fma(Float64(Float64(-4.0 * z) * a), t, b)) / Float64(z * c));
	else
		tmp = fma(Float64(Float64(x / Float64(z * c)) * 9.0), y, fma(Float64(Float64(a / c) * t), -4.0, Float64(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_] := If[LessEqual[z, -2.4e-67], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0 + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[z, 1.75e-20], N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * y + N[(N[(N[(a / c), $MachinePrecision] * t), $MachinePrecision] * -4.0 + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e-67

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 97.3%

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

   if -2.4e-67 < z < 1.75000000000000002e-20

   1. Initial program 97.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   4. Applied rewrites 97.4%

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

   if 1.75000000000000002e-20 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 87.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \mathsf{fma}\left(\frac{y \cdot x}{z}, 9, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z \cdot c} \cdot 9, y, \mathsf{fma}\left(\frac{a}{c} \cdot t, -4, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\left(\left(9 \cdot x\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_2 := \frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \end{array} \]

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 (/ (+ (- (* (* 9.0 x) y) (* (* (* 4.0 z) t) a)) b) (* z c)))
        (t_2 (/ (fma (* y 9.0) x (fma (* (* -4.0 z) a) t b)) (* z c))))
   (if (<= t_1 -4e-80)
     t_2
     (if (<= t_1 0.0)
       (/ (fma (* -4.0 a) t (/ b z)) c)
       (if (<= t_1 INFINITY) t_2 (/ (* -4.0 a) (/ c t)))))))

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 = ((((9.0 * x) * y) - (((4.0 * z) * t) * a)) + b) / (z * c);
	double t_2 = fma((y * 9.0), x, fma(((-4.0 * z) * a), t, b)) / (z * c);
	double tmp;
	if (t_1 <= -4e-80) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (-4.0 * a) / (c / t);
	}
	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(Float64(Float64(Float64(9.0 * x) * y) - Float64(Float64(Float64(4.0 * z) * t) * a)) + b) / Float64(z * c))
	t_2 = Float64(fma(Float64(y * 9.0), x, fma(Float64(Float64(-4.0 * z) * a), t, b)) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -4e-80)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c / t));
	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[(N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-80], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(-4.0 * a), $MachinePrecision] / N[(c / t), $MachinePrecision]), $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)) < -3.99999999999999985e-80 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   4. Applied rewrites 90.8%

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

   if -3.99999999999999985e-80 < (/.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 62.2%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 99.7%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 86.6%

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

   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. 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. lower-*.f64 9.5

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

   5. Applied rewrites 9.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 61.2

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 61.2%

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

   10. Applied rewrites 86.9%

       \[\leadsto \frac{a \cdot -4}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(9 \cdot x\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(9 \cdot x\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(9 \cdot x\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ t_2 := \frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{y \cdot x}{z} \cdot 9\right)}{c}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-83}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y))
        (t_2 (/ (fma (* -4.0 t) a (* (/ (* y x) z) 9.0)) c)))
   (if (<= t_1 -2e-83)
     t_2
     (if (<= t_1 1e-268)
       (/ (fma (* -4.0 a) t (/ b z)) c)
       (if (<= t_1 5e+55) (fma (* -4.0 a) (/ t c) (/ b (* z c))) t_2)))))

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 = (9.0 * x) * y;
	double t_2 = fma((-4.0 * t), a, (((y * x) / z) * 9.0)) / c;
	double tmp;
	if (t_1 <= -2e-83) {
		tmp = t_2;
	} else if (t_1 <= 1e-268) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else if (t_1 <= 5e+55) {
		tmp = fma((-4.0 * a), (t / c), (b / (z * c)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(9.0 * x) * y)
	t_2 = Float64(fma(Float64(-4.0 * t), a, Float64(Float64(Float64(y * x) / z) * 9.0)) / c)
	tmp = 0.0
	if (t_1 <= -2e-83)
		tmp = t_2;
	elseif (t_1 <= 1e-268)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	elseif (t_1 <= 5e+55)
		tmp = fma(Float64(-4.0 * a), Float64(t / c), Float64(b / Float64(z * c)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-83], t$95$2, If[LessEqual[t$95$1, 1e-268], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 5e+55], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.0000000000000001e-83 or 5.00000000000000046e55 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 79.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.0%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 84.6%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 78.5%

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

   if -2.0000000000000001e-83 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999958e-269

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.6%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 94.3%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 91.9%

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

   if 9.99999999999999958e-269 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000046e55

   1. Initial program 77.7%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{y \cdot x}{z} \cdot 9\right)}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{y \cdot x}{z} \cdot 9\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot t\right) \cdot z, a, \left(y \cdot x\right) \cdot 9\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)))
   (if (<= t_1 -4e+121)
     (/ (fma (* (* -4.0 t) z) a (* (* y x) 9.0)) (* z c))
     (if (<= t_1 1e-268)
       (/ (fma (* -4.0 a) t (/ b z)) c)
       (if (<= t_1 1e+99)
         (fma (* -4.0 a) (/ t c) (/ b (* z c)))
         (/ (/ (fma (* y x) 9.0 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 = (9.0 * x) * y;
	double tmp;
	if (t_1 <= -4e+121) {
		tmp = fma(((-4.0 * t) * z), a, ((y * x) * 9.0)) / (z * c);
	} else if (t_1 <= 1e-268) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else if (t_1 <= 1e+99) {
		tmp = fma((-4.0 * a), (t / c), (b / (z * c)));
	} else {
		tmp = (fma((y * x), 9.0, 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(9.0 * x) * y)
	tmp = 0.0
	if (t_1 <= -4e+121)
		tmp = Float64(fma(Float64(Float64(-4.0 * t) * z), a, Float64(Float64(y * x) * 9.0)) / Float64(z * c));
	elseif (t_1 <= 1e-268)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	elseif (t_1 <= 1e+99)
		tmp = fma(Float64(-4.0 * a), Float64(t / c), Float64(b / Float64(z * c)));
	else
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, 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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+121], N[(N[(N[(N[(-4.0 * t), $MachinePrecision] * z), $MachinePrecision] * a + N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-268], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 1e+99], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121

   1. Initial program 87.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. associate-*r/ N/A

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

      5. div-sub N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.2%

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

   9. Taylor expanded in b around 0

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

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 83.7

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

   11. Applied rewrites 83.7%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999958e-269

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 92.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 84.0%

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

   if 9.99999999999999958e-269 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.2%

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

   if 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot t\right) \cdot z, a, \left(y \cdot x\right) \cdot 9\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, \left(\left(t \cdot z\right) \cdot a\right) \cdot -4\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)))
   (if (<= t_1 -4e+121)
     (/ (fma (* y x) 9.0 (* (* (* t z) a) -4.0)) (* z c))
     (if (<= t_1 1e-268)
       (/ (fma (* -4.0 a) t (/ b z)) c)
       (if (<= t_1 1e+99)
         (fma (* -4.0 a) (/ t c) (/ b (* z c)))
         (/ (/ (fma (* y x) 9.0 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 = (9.0 * x) * y;
	double tmp;
	if (t_1 <= -4e+121) {
		tmp = fma((y * x), 9.0, (((t * z) * a) * -4.0)) / (z * c);
	} else if (t_1 <= 1e-268) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else if (t_1 <= 1e+99) {
		tmp = fma((-4.0 * a), (t / c), (b / (z * c)));
	} else {
		tmp = (fma((y * x), 9.0, 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(9.0 * x) * y)
	tmp = 0.0
	if (t_1 <= -4e+121)
		tmp = Float64(fma(Float64(y * x), 9.0, Float64(Float64(Float64(t * z) * a) * -4.0)) / Float64(z * c));
	elseif (t_1 <= 1e-268)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	elseif (t_1 <= 1e+99)
		tmp = fma(Float64(-4.0 * a), Float64(t / c), Float64(b / Float64(z * c)));
	else
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, 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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+121], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + N[(N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-268], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 1e+99], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121

   1. Initial program 87.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. associate-*r/ N/A

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

      5. div-sub N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   7. 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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.6

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

   8. Applied rewrites 83.6%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999958e-269

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 92.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 84.0%

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

   if 9.99999999999999958e-269 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.2%

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

   if 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, \left(\left(t \cdot z\right) \cdot a\right) \cdot -4\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{\mathsf{fma}\left(\frac{x}{c} \cdot -9, y, \frac{-b}{c}\right)}{-z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c 4e-20)
   (/ (- (/ (fma (* y x) 9.0 b) z) (* (* a t) 4.0)) c)
   (fma (* -4.0 a) (/ t c) (/ (fma (* (/ x c) -9.0) y (/ (- b) c)) (- z)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= 4e-20) {
		tmp = ((fma((y * x), 9.0, b) / z) - ((a * t) * 4.0)) / c;
	} else {
		tmp = fma((-4.0 * a), (t / c), (fma(((x / c) * -9.0), y, (-b / c)) / -z));
	}
	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)
	tmp = 0.0
	if (c <= 4e-20)
		tmp = Float64(Float64(Float64(fma(Float64(y * x), 9.0, b) / z) - Float64(Float64(a * t) * 4.0)) / c);
	else
		tmp = fma(Float64(-4.0 * a), Float64(t / c), Float64(fma(Float64(Float64(x / c) * -9.0), y, Float64(Float64(-b) / c)) / Float64(-z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, 4e-20], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] - N[(N[(a * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(N[(N[(N[(x / c), $MachinePrecision] * -9.0), $MachinePrecision] * y + N[((-b) / c), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 3.99999999999999978e-20

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 91.7%

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

   if 3.99999999999999978e-20 < c

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 94.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{\mathsf{fma}\left(\frac{x}{c} \cdot -9, y, \frac{-b}{c}\right)}{-z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ t_2 := \left(9 \cdot x\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{t\_1}{z \cdot c}\\ \mathbf{elif}\;t\_2 \leq 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{z}}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b)) (t_2 (* (* 9.0 x) y)))
   (if (<= t_2 -4e+121)
     (/ t_1 (* z c))
     (if (<= t_2 1e+99) (/ (fma (* -4.0 a) t (/ b z)) c) (/ (/ t_1 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 = fma((y * x), 9.0, b);
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -4e+121) {
		tmp = t_1 / (z * c);
	} else if (t_2 <= 1e+99) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else {
		tmp = (t_1 / 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 = fma(Float64(y * x), 9.0, b)
	t_2 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_2 <= -4e+121)
		tmp = Float64(t_1 / Float64(z * c));
	elseif (t_2 <= 1e+99)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	else
		tmp = Float64(Float64(t_1 / 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[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+121], N[(t$95$1 / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+99], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(t$95$1 / z), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 87.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 79.1%

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

   if 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ t_2 := \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)) (t_2 (/ (fma (* y x) 9.0 b) (* z c))))
   (if (<= t_1 -4e+121)
     t_2
     (if (<= t_1 1e+99) (/ (fma (* -4.0 a) t (/ b z)) c) t_2))))

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 = (9.0 * x) * y;
	double t_2 = fma((y * x), 9.0, b) / (z * c);
	double tmp;
	if (t_1 <= -4e+121) {
		tmp = t_2;
	} else if (t_1 <= 1e+99) {
		tmp = fma((-4.0 * a), t, (b / z)) / c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(9.0 * x) * y)
	t_2 = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -4e+121)
		tmp = t_2;
	elseif (t_1 <= 1e+99)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+121], t$95$2, If[LessEqual[t$95$1, 1e+99], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121 or 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 82.2%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 87.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{\frac{b}{c \cdot z}} \]
   13. Step-by-step derivation

   14. Applied rewrites 79.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \mathsf{fma}\left(\frac{y \cdot x}{z}, 9, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \end{array} \end{array} \]

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
 (if (<= z -2.4e-67)
   (/ (fma (* a t) -4.0 (fma (/ (* y x) z) 9.0 (/ b z))) c)
   (if (<= z 2.5e+16)
     (/ (fma (* y 9.0) x (fma (* (* -4.0 z) a) t b)) (* z c))
     (/ (- (/ (fma (* y x) 9.0 b) z) (* (* a t) 4.0)) 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 tmp;
	if (z <= -2.4e-67) {
		tmp = fma((a * t), -4.0, fma(((y * x) / z), 9.0, (b / z))) / c;
	} else if (z <= 2.5e+16) {
		tmp = fma((y * 9.0), x, fma(((-4.0 * z) * a), t, b)) / (z * c);
	} else {
		tmp = ((fma((y * x), 9.0, b) / z) - ((a * t) * 4.0)) / 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)
	tmp = 0.0
	if (z <= -2.4e-67)
		tmp = Float64(fma(Float64(a * t), -4.0, fma(Float64(Float64(y * x) / z), 9.0, Float64(b / z))) / c);
	elseif (z <= 2.5e+16)
		tmp = Float64(fma(Float64(y * 9.0), x, fma(Float64(Float64(-4.0 * z) * a), t, b)) / Float64(z * c));
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * x), 9.0, b) / z) - Float64(Float64(a * t) * 4.0)) / c);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[z, -2.4e-67], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0 + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[z, 2.5e+16], N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] - N[(N[(a * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e-67

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 97.3%

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

   if -2.4e-67 < z < 2.5e16

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

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

   if 2.5e16 < z

   1. Initial program 46.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 82.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \mathsf{fma}\left(\frac{y \cdot x}{z}, 9, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot c}\\ t_2 := \left(9 \cdot x\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot 9\\ \mathbf{elif}\;t\_2 \leq 10^{+99}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 9\right) \cdot t\_1\\ \end{array} \end{array} \]

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 (/ x (* z c))) (t_2 (* (* 9.0 x) y)))
   (if (<= t_2 -4e+121)
     (* (* t_1 y) 9.0)
     (if (<= t_2 1e+99) (/ (* -4.0 a) (/ c t)) (* (* y 9.0) 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 = x / (z * c);
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -4e+121) {
		tmp = (t_1 * y) * 9.0;
	} else if (t_2 <= 1e+99) {
		tmp = (-4.0 * a) / (c / t);
	} else {
		tmp = (y * 9.0) * 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) :: t_2
    real(8) :: tmp
    t_1 = x / (z * c)
    t_2 = (9.0d0 * x) * y
    if (t_2 <= (-4d+121)) then
        tmp = (t_1 * y) * 9.0d0
    else if (t_2 <= 1d+99) then
        tmp = ((-4.0d0) * a) / (c / t)
    else
        tmp = (y * 9.0d0) * 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 = x / (z * c);
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -4e+121) {
		tmp = (t_1 * y) * 9.0;
	} else if (t_2 <= 1e+99) {
		tmp = (-4.0 * a) / (c / t);
	} else {
		tmp = (y * 9.0) * 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 = x / (z * c)
	t_2 = (9.0 * x) * y
	tmp = 0
	if t_2 <= -4e+121:
		tmp = (t_1 * y) * 9.0
	elif t_2 <= 1e+99:
		tmp = (-4.0 * a) / (c / t)
	else:
		tmp = (y * 9.0) * 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(x / Float64(z * c))
	t_2 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_2 <= -4e+121)
		tmp = Float64(Float64(t_1 * y) * 9.0);
	elseif (t_2 <= 1e+99)
		tmp = Float64(Float64(-4.0 * a) / Float64(c / t));
	else
		tmp = Float64(Float64(y * 9.0) * 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 = x / (z * c);
	t_2 = (9.0 * x) * y;
	tmp = 0.0;
	if (t_2 <= -4e+121)
		tmp = (t_1 * y) * 9.0;
	elseif (t_2 <= 1e+99)
		tmp = (-4.0 * a) / (c / t);
	else
		tmp = (y * 9.0) * 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[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+121], N[(N[(t$95$1 * y), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+99], N[(N[(-4.0 * a), $MachinePrecision] / N[(c / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 9.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.5%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.1%

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

   3. Taylor expanded in 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. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 47.9

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 52.0%

       \[\leadsto \frac{a \cdot -4}{\color{blue}{\frac{c}{t}}} \]

   if 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 76.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 9\right) \cdot \frac{x}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* y x) 9.0 b) z) (* (* a t) 4.0)) c)))
   (if (<= z -2.4e-67)
     t_1
     (if (<= z 2.5e+16)
       (/ (fma (* y 9.0) x (fma (* (* -4.0 z) a) 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((y * x), 9.0, b) / z) - ((a * t) * 4.0)) / c;
	double tmp;
	if (z <= -2.4e-67) {
		tmp = t_1;
	} else if (z <= 2.5e+16) {
		tmp = fma((y * 9.0), x, fma(((-4.0 * z) * a), 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(Float64(Float64(fma(Float64(y * x), 9.0, b) / z) - Float64(Float64(a * t) * 4.0)) / c)
	tmp = 0.0
	if (z <= -2.4e-67)
		tmp = t_1;
	elseif (z <= 2.5e+16)
		tmp = Float64(fma(Float64(y * 9.0), x, fma(Float64(Float64(-4.0 * z) * a), 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[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] - N[(N[(a * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[z, -2.4e-67], t$95$1, If[LessEqual[z, 2.5e+16], N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e-67 or 2.5e16 < z

   1. Initial program 63.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 91.1%

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

   if -2.4e-67 < z < 2.5e16

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z} - \left(a \cdot t\right) \cdot 4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot c}\\ t_2 := \left(9 \cdot x\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot 9\\ \mathbf{elif}\;t\_2 \leq 10^{+99}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 9\right) \cdot t\_1\\ \end{array} \end{array} \]

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 (/ x (* z c))) (t_2 (* (* 9.0 x) y)))
   (if (<= t_2 -4e+121)
     (* (* t_1 y) 9.0)
     (if (<= t_2 1e+99) (* (/ t c) (* -4.0 a)) (* (* y 9.0) 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 = x / (z * c);
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -4e+121) {
		tmp = (t_1 * y) * 9.0;
	} else if (t_2 <= 1e+99) {
		tmp = (t / c) * (-4.0 * a);
	} else {
		tmp = (y * 9.0) * 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) :: t_2
    real(8) :: tmp
    t_1 = x / (z * c)
    t_2 = (9.0d0 * x) * y
    if (t_2 <= (-4d+121)) then
        tmp = (t_1 * y) * 9.0d0
    else if (t_2 <= 1d+99) then
        tmp = (t / c) * ((-4.0d0) * a)
    else
        tmp = (y * 9.0d0) * 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 = x / (z * c);
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -4e+121) {
		tmp = (t_1 * y) * 9.0;
	} else if (t_2 <= 1e+99) {
		tmp = (t / c) * (-4.0 * a);
	} else {
		tmp = (y * 9.0) * 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 = x / (z * c)
	t_2 = (9.0 * x) * y
	tmp = 0
	if t_2 <= -4e+121:
		tmp = (t_1 * y) * 9.0
	elif t_2 <= 1e+99:
		tmp = (t / c) * (-4.0 * a)
	else:
		tmp = (y * 9.0) * 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(x / Float64(z * c))
	t_2 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_2 <= -4e+121)
		tmp = Float64(Float64(t_1 * y) * 9.0);
	elseif (t_2 <= 1e+99)
		tmp = Float64(Float64(t / c) * Float64(-4.0 * a));
	else
		tmp = Float64(Float64(y * 9.0) * 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 = x / (z * c);
	t_2 = (9.0 * x) * y;
	tmp = 0.0;
	if (t_2 <= -4e+121)
		tmp = (t_1 * y) * 9.0;
	elseif (t_2 <= 1e+99)
		tmp = (t / c) * (-4.0 * a);
	else
		tmp = (y * 9.0) * 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[(x / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+121], N[(N[(t$95$1 * y), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+99], N[(N[(t / c), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(y * 9.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121

   1. Initial program 87.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.5%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.1%

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

   3. Taylor expanded in 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. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 47.9

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 52.0%

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

   if 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 76.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 9\right) \cdot \frac{x}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ t_2 := \left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+99}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)) (t_2 (* (* (/ x (* z c)) y) 9.0)))
   (if (<= t_1 -4e+121) t_2 (if (<= t_1 1e+99) (* (/ t c) (* -4.0 a)) t_2))))

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 = (9.0 * x) * y;
	double t_2 = ((x / (z * c)) * y) * 9.0;
	double tmp;
	if (t_1 <= -4e+121) {
		tmp = t_2;
	} else if (t_1 <= 1e+99) {
		tmp = (t / c) * (-4.0 * a);
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = (9.0d0 * x) * y
    t_2 = ((x / (z * c)) * y) * 9.0d0
    if (t_1 <= (-4d+121)) then
        tmp = t_2
    else if (t_1 <= 1d+99) then
        tmp = (t / c) * ((-4.0d0) * a)
    else
        tmp = t_2
    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 = (9.0 * x) * y;
	double t_2 = ((x / (z * c)) * y) * 9.0;
	double tmp;
	if (t_1 <= -4e+121) {
		tmp = t_2;
	} else if (t_1 <= 1e+99) {
		tmp = (t / c) * (-4.0 * a);
	} else {
		tmp = t_2;
	}
	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 = (9.0 * x) * y
	t_2 = ((x / (z * c)) * y) * 9.0
	tmp = 0
	if t_1 <= -4e+121:
		tmp = t_2
	elif t_1 <= 1e+99:
		tmp = (t / c) * (-4.0 * a)
	else:
		tmp = t_2
	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(9.0 * x) * y)
	t_2 = Float64(Float64(Float64(x / Float64(z * c)) * y) * 9.0)
	tmp = 0.0
	if (t_1 <= -4e+121)
		tmp = t_2;
	elseif (t_1 <= 1e+99)
		tmp = Float64(Float64(t / c) * Float64(-4.0 * a));
	else
		tmp = t_2;
	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 = (9.0 * x) * y;
	t_2 = ((x / (z * c)) * y) * 9.0;
	tmp = 0.0;
	if (t_1 <= -4e+121)
		tmp = t_2;
	elseif (t_1 <= 1e+99)
		tmp = (t / c) * (-4.0 * a);
	else
		tmp = t_2;
	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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / N[(z * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 9.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+121], t$95$2, If[LessEqual[t$95$1, 1e+99], N[(N[(t / c), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.00000000000000015e121 or 9.9999999999999997e98 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

   if -4.00000000000000015e121 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e98

   1. Initial program 76.1%

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

   3. Taylor expanded in 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. lower-*.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 47.9

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 52.0%

       \[\leadsto \frac{t}{c} \cdot \color{blue}{\left(a \cdot -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10^{+99}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.82 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= a -1.82e-69)
   (/ (fma -4.0 (* (* t z) a) b) (* z c))
   (if (<= a 5.4e+76)
     (/ (/ (fma (* y x) 9.0 b) z) c)
     (fma (* -4.0 a) (/ t c) (/ 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 tmp;
	if (a <= -1.82e-69) {
		tmp = fma(-4.0, ((t * z) * a), b) / (z * c);
	} else if (a <= 5.4e+76) {
		tmp = (fma((y * x), 9.0, b) / z) / c;
	} else {
		tmp = fma((-4.0 * a), (t / c), (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)
	tmp = 0.0
	if (a <= -1.82e-69)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(z * c));
	elseif (a <= 5.4e+76)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / z) / c);
	else
		tmp = fma(Float64(-4.0 * a), Float64(t / c), Float64(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_] := If[LessEqual[a, -1.82e-69], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.4e+76], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision] + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.82e-69

   1. Initial program 87.8%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if -1.82e-69 < a < 5.3999999999999998e76

   1. Initial program 77.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 77.0%

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

   if 5.3999999999999998e76 < a

   1. Initial program 64.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{c \cdot z} \cdot 9, y, \mathsf{fma}\left(\frac{a \cdot t}{c}, -4, \frac{b}{c \cdot z}\right)\right)} \]

   6. Taylor expanded in z around -inf

      \[\leadsto -4 \cdot \frac{a \cdot t}{c} + \color{blue}{-1 \cdot \frac{-9 \cdot \frac{x \cdot y}{c} + -1 \cdot \frac{b}{c}}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(-4 \cdot a, \color{blue}{\frac{t}{c}}, \frac{\mathsf{fma}\left(\frac{x}{c} \cdot -9, y, \frac{-b}{c}\right)}{-z}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{c \cdot z}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.3%

       \[\leadsto \mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.82 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot a, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \end{array} \]

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
 (if (<= t -6.2e+187)
   (/ (fma -4.0 (* (* t z) a) b) (* z c))
   (if (<= t 3.3e-40) (/ (fma (* y x) 9.0 b) (* z c)) (/ (* -4.0 a) (/ c t)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -6.2e+187) {
		tmp = fma(-4.0, ((t * z) * a), b) / (z * c);
	} else if (t <= 3.3e-40) {
		tmp = fma((y * x), 9.0, b) / (z * c);
	} else {
		tmp = (-4.0 * a) / (c / t);
	}
	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)
	tmp = 0.0
	if (t <= -6.2e+187)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(z * c));
	elseif (t <= 3.3e-40)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c));
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c / t));
	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_] := If[LessEqual[t, -6.2e+187], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-40], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] / N[(c / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.20000000000000024e187

   1. Initial program 70.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      7. lower-*.f64 66.1

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right)} \cdot a, b\right)}{z \cdot c} \]

   5. Applied rewrites 66.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}}{z \cdot c} \]

   if -6.20000000000000024e187 < t < 3.29999999999999993e-40

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z \cdot c} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z \cdot c} \]

      5. lower-*.f64 71.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z \cdot c} \]

   5. Applied rewrites 71.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot x, 9, b\right)}}{z \cdot c} \]

   if 3.29999999999999993e-40 < t

   1. Initial program 68.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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. lower-*.f64 27.8

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 44.4

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.7%

       \[\leadsto \frac{a \cdot -4}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{-4 \cdot t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \end{array} \]

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
 (if (<= t -4.2e+183)
   (/ (* -4.0 t) (/ c a))
   (if (<= t 3.3e-40) (/ (fma (* y x) 9.0 b) (* z c)) (/ (* -4.0 a) (/ c t)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -4.2e+183) {
		tmp = (-4.0 * t) / (c / a);
	} else if (t <= 3.3e-40) {
		tmp = fma((y * x), 9.0, b) / (z * c);
	} else {
		tmp = (-4.0 * a) / (c / t);
	}
	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)
	tmp = 0.0
	if (t <= -4.2e+183)
		tmp = Float64(Float64(-4.0 * t) / Float64(c / a));
	elseif (t <= 3.3e-40)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c));
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c / t));
	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_] := If[LessEqual[t, -4.2e+183], N[(N[(-4.0 * t), $MachinePrecision] / N[(c / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-40], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] / N[(c / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2e183

   1. Initial program 70.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 21.4

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 66.0

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.3%

       \[\leadsto \left(-4 \cdot t\right) \cdot \color{blue}{\frac{a}{c}} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.4%

       \[\leadsto \frac{-4 \cdot t}{\color{blue}{\frac{c}{a}}} \]

   if -4.2e183 < t < 3.29999999999999993e-40

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z \cdot c} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z \cdot c} \]

      5. lower-*.f64 71.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z \cdot c} \]

   5. Applied rewrites 71.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot x, 9, b\right)}}{z \cdot c} \]

   if 3.29999999999999993e-40 < t

   1. Initial program 68.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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. lower-*.f64 27.8

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 44.4

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.7%

       \[\leadsto \frac{a \cdot -4}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{-4 \cdot t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot t\right) \cdot \frac{a}{c}\\ \end{array} \end{array} \]

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
 (if (<= t -5.5e-39)
   (* (/ t c) (* -4.0 a))
   (if (<= t 3.8e-183) (/ b (* z c)) (* (* -4.0 t) (/ a 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 tmp;
	if (t <= -5.5e-39) {
		tmp = (t / c) * (-4.0 * a);
	} else if (t <= 3.8e-183) {
		tmp = b / (z * c);
	} else {
		tmp = (-4.0 * t) * (a / c);
	}
	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) :: tmp
    if (t <= (-5.5d-39)) then
        tmp = (t / c) * ((-4.0d0) * a)
    else if (t <= 3.8d-183) then
        tmp = b / (z * c)
    else
        tmp = ((-4.0d0) * t) * (a / c)
    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 tmp;
	if (t <= -5.5e-39) {
		tmp = (t / c) * (-4.0 * a);
	} else if (t <= 3.8e-183) {
		tmp = b / (z * c);
	} else {
		tmp = (-4.0 * t) * (a / c);
	}
	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):
	tmp = 0
	if t <= -5.5e-39:
		tmp = (t / c) * (-4.0 * a)
	elif t <= 3.8e-183:
		tmp = b / (z * c)
	else:
		tmp = (-4.0 * t) * (a / 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)
	tmp = 0.0
	if (t <= -5.5e-39)
		tmp = Float64(Float64(t / c) * Float64(-4.0 * a));
	elseif (t <= 3.8e-183)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = Float64(Float64(-4.0 * t) * Float64(a / c));
	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)
	tmp = 0.0;
	if (t <= -5.5e-39)
		tmp = (t / c) * (-4.0 * a);
	elseif (t <= 3.8e-183)
		tmp = b / (z * c);
	else
		tmp = (-4.0 * t) * (a / c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -5.5e-39], N[(N[(t / c), $MachinePrecision] * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-183], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * t), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.50000000000000018e-39

   1. Initial program 76.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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. lower-*.f64 27.9

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 46.8

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.4%

       \[\leadsto \frac{t}{c} \cdot \color{blue}{\left(a \cdot -4\right)} \]

   if -5.50000000000000018e-39 < t < 3.7999999999999996e-183

   1. Initial program 88.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. lower-*.f64 40.6

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   if 3.7999999999999996e-183 < t

   1. Initial program 71.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 29.7

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 42.4

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.9%

       \[\leadsto \left(-4 \cdot t\right) \cdot \color{blue}{\frac{a}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{t}{c} \cdot \left(-4 \cdot a\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot t\right) \cdot \frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(-4 \cdot t\right) \cdot \frac{a}{c}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (<= t -5.5e-39) t_1 (if (<= t 3.8e-183) (/ 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 (t <= -5.5e-39) {
		tmp = t_1;
	} else if (t <= 3.8e-183) {
		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 (t <= (-5.5d-39)) then
        tmp = t_1
    else if (t <= 3.8d-183) 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 (t <= -5.5e-39) {
		tmp = t_1;
	} else if (t <= 3.8e-183) {
		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 t <= -5.5e-39:
		tmp = t_1
	elif t <= 3.8e-183:
		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 * t) * Float64(a / c))
	tmp = 0.0
	if (t <= -5.5e-39)
		tmp = t_1;
	elseif (t <= 3.8e-183)
		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 (t <= -5.5e-39)
		tmp = t_1;
	elseif (t <= 3.8e-183)
		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 * t), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.5e-39], t$95$1, If[LessEqual[t, 3.8e-183], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.50000000000000018e-39 or 3.7999999999999996e-183 < t

   1. Initial program 73.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 29.0

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c}} \cdot -4 \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

      5. lower-*.f64 44.1

         \[\leadsto \frac{\color{blue}{t \cdot a}}{c} \cdot -4 \]

   8. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{t \cdot a}{c} \cdot -4} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.6%

       \[\leadsto \left(-4 \cdot t\right) \cdot \color{blue}{\frac{a}{c}} \]

   if -5.50000000000000018e-39 < t < 3.7999999999999996e-183

   1. Initial program 88.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. lower-*.f64 40.6

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\left(-4 \cdot t\right) \cdot \frac{a}{c}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot t\right) \cdot \frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.3% 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 78.3%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   2. lower-*.f64 32.7

      \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

5. Applied rewrites 32.7%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

6. Final simplification 32.7%

   \[\leadsto \frac{b}{z \cdot c} \]
7. Add Preprocessing

Developer Target 1: 80.4% 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 2024332 
(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)))