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

Percentage Accurate: 79.6% → 90.0%

Time: 13.2s

Alternatives: 21

Speedup: 0.8×

• 38.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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)) < +inf.0

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 92.9

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

   7. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 92.9%

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

   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 a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

4. Final simplification 93.1%

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

Alternative 2: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_1 -1e-283)
     t_1
     (if (<= t_1 0.0)
       (/ (/ (fma (* y x) 9.0 b) c) z)
       (if (<= t_1 INFINITY)
         (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) (* z c))
         (* (* -4.0 a) (/ t c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_1 <= -1e-283) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (fma((y * x), 9.0, b) / c) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / (z * c);
	} else {
		tmp = (-4.0 * a) * (t / c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e-283)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, b)) / Float64(z * c));
	else
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(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]}, If[LessEqual[t$95$1, -1e-283], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -9.99999999999999947e-284

   1. Initial program 92.4%

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

   if -9.99999999999999947e-284 < (/.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 28.4%

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

   3. Taylor expanded in z around 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 67.2

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

   5. Applied rewrites 67.2%

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

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

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 89.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 54.0%

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

Alternative 3: 87.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4 \cdot z, y \cdot \left(9 \cdot x\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_1 -1e-283)
     (/ (+ (fma (* a t) (* -4.0 z) (* y (* 9.0 x))) b) (* z c))
     (if (<= t_1 0.0)
       (/ (/ (fma (* y x) 9.0 b) c) z)
       (if (<= t_1 INFINITY)
         (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) (* z c))
         (* (* -4.0 a) (/ t c)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_1 <= -1e-283) {
		tmp = (fma((a * t), (-4.0 * z), (y * (9.0 * x))) + b) / (z * c);
	} else if (t_1 <= 0.0) {
		tmp = (fma((y * x), 9.0, b) / c) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / (z * c);
	} else {
		tmp = (-4.0 * a) * (t / c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
	tmp = 0.0
	if (t_1 <= -1e-283)
		tmp = Float64(Float64(fma(Float64(a * t), Float64(-4.0 * z), Float64(y * Float64(9.0 * x))) + b) / Float64(z * c));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, b)) / Float64(z * c));
	else
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(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]}, If[LessEqual[t$95$1, -1e-283], N[(N[(N[(N[(a * t), $MachinePrecision] * N[(-4.0 * z), $MachinePrecision] + N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -9.99999999999999947e-284

   1. Initial program 92.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 91.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 91.4

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if -9.99999999999999947e-284 < (/.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 28.4%

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

   3. Taylor expanded in z around 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 67.2

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

   5. Applied rewrites 67.2%

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

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

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 89.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 54.0%

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

Alternative 4: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := \left(x \cdot 9\right) \cdot y\\ t_3 := \left(9 \cdot x\right) \cdot \frac{y}{c \cdot z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+34}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{a \cdot -4}{\frac{c}{t}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (/ b c) z))
        (t_2 (* (* x 9.0) y))
        (t_3 (* (* 9.0 x) (/ y (* c z)))))
   (if (<= t_2 -5e+104)
     t_3
     (if (<= t_2 -4e+34)
       (* (* -4.0 a) (/ t c))
       (if (<= t_2 -4e-137)
         t_1
         (if (<= t_2 5e-58)
           (/ (* a -4.0) (/ c t))
           (if (<= t_2 5e+15) t_1 t_3)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b / c) / z;
	double t_2 = (x * 9.0) * y;
	double t_3 = (9.0 * x) * (y / (c * z));
	double tmp;
	if (t_2 <= -5e+104) {
		tmp = t_3;
	} else if (t_2 <= -4e+34) {
		tmp = (-4.0 * a) * (t / c);
	} else if (t_2 <= -4e-137) {
		tmp = t_1;
	} else if (t_2 <= 5e-58) {
		tmp = (a * -4.0) / (c / t);
	} else if (t_2 <= 5e+15) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (b / c) / z
    t_2 = (x * 9.0d0) * y
    t_3 = (9.0d0 * x) * (y / (c * z))
    if (t_2 <= (-5d+104)) then
        tmp = t_3
    else if (t_2 <= (-4d+34)) then
        tmp = ((-4.0d0) * a) * (t / c)
    else if (t_2 <= (-4d-137)) then
        tmp = t_1
    else if (t_2 <= 5d-58) then
        tmp = (a * (-4.0d0)) / (c / t)
    else if (t_2 <= 5d+15) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
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 = (b / c) / z;
	double t_2 = (x * 9.0) * y;
	double t_3 = (9.0 * x) * (y / (c * z));
	double tmp;
	if (t_2 <= -5e+104) {
		tmp = t_3;
	} else if (t_2 <= -4e+34) {
		tmp = (-4.0 * a) * (t / c);
	} else if (t_2 <= -4e-137) {
		tmp = t_1;
	} else if (t_2 <= 5e-58) {
		tmp = (a * -4.0) / (c / t);
	} else if (t_2 <= 5e+15) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[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 = (b / c) / z
	t_2 = (x * 9.0) * y
	t_3 = (9.0 * x) * (y / (c * z))
	tmp = 0
	if t_2 <= -5e+104:
		tmp = t_3
	elif t_2 <= -4e+34:
		tmp = (-4.0 * a) * (t / c)
	elif t_2 <= -4e-137:
		tmp = t_1
	elif t_2 <= 5e-58:
		tmp = (a * -4.0) / (c / t)
	elif t_2 <= 5e+15:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b / c) / z)
	t_2 = Float64(Float64(x * 9.0) * y)
	t_3 = Float64(Float64(9.0 * x) * Float64(y / Float64(c * z)))
	tmp = 0.0
	if (t_2 <= -5e+104)
		tmp = t_3;
	elseif (t_2 <= -4e+34)
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	elseif (t_2 <= -4e-137)
		tmp = t_1;
	elseif (t_2 <= 5e-58)
		tmp = Float64(Float64(a * -4.0) / Float64(c / t));
	elseif (t_2 <= 5e+15)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
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 = (b / c) / z;
	t_2 = (x * 9.0) * y;
	t_3 = (9.0 * x) * (y / (c * z));
	tmp = 0.0;
	if (t_2 <= -5e+104)
		tmp = t_3;
	elseif (t_2 <= -4e+34)
		tmp = (-4.0 * a) * (t / c);
	elseif (t_2 <= -4e-137)
		tmp = t_1;
	elseif (t_2 <= 5e-58)
		tmp = (a * -4.0) / (c / t);
	elseif (t_2 <= 5e+15)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(9.0 * x), $MachinePrecision] * N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+104], t$95$3, If[LessEqual[t$95$2, -4e+34], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-137], t$95$1, If[LessEqual[t$95$2, 5e-58], N[(N[(a * -4.0), $MachinePrecision] / N[(c / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+15], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104 or 5e15 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   10. Applied rewrites 73.3%

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

   12. Applied rewrites 69.6%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999978e34

   1. Initial program 88.3%

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

   3. Taylor expanded in z around inf

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

      1. *-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. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 68.1%

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

   if -3.99999999999999978e34 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999991e-137 or 4.99999999999999977e-58 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5e15

   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 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 66.2%

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

   if -3.99999999999999991e-137 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999977e-58

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 52.4%

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

   9. Applied rewrites 52.4%

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

Alternative 5: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := \left(-4 \cdot a\right) \cdot \frac{t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(9 \cdot x\right) \cdot \frac{y}{c \cdot z}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (/ b c) z))
        (t_2 (* (* -4.0 a) (/ t c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (* (* 9.0 x) (/ y (* c z)))))
   (if (<= t_3 -5e+104)
     t_4
     (if (<= t_3 -4e+34)
       t_2
       (if (<= t_3 -4e-137)
         t_1
         (if (<= t_3 5e-58) t_2 (if (<= t_3 5e+15) t_1 t_4)))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b / c) / z;
	double t_2 = (-4.0 * a) * (t / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (9.0 * x) * (y / (c * z));
	double tmp;
	if (t_3 <= -5e+104) {
		tmp = t_4;
	} else if (t_3 <= -4e+34) {
		tmp = t_2;
	} else if (t_3 <= -4e-137) {
		tmp = t_1;
	} else if (t_3 <= 5e-58) {
		tmp = t_2;
	} else if (t_3 <= 5e+15) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (b / c) / z
    t_2 = ((-4.0d0) * a) * (t / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (9.0d0 * x) * (y / (c * z))
    if (t_3 <= (-5d+104)) then
        tmp = t_4
    else if (t_3 <= (-4d+34)) then
        tmp = t_2
    else if (t_3 <= (-4d-137)) then
        tmp = t_1
    else if (t_3 <= 5d-58) then
        tmp = t_2
    else if (t_3 <= 5d+15) then
        tmp = t_1
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (b / c) / z;
	double t_2 = (-4.0 * a) * (t / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (9.0 * x) * (y / (c * z));
	double tmp;
	if (t_3 <= -5e+104) {
		tmp = t_4;
	} else if (t_3 <= -4e+34) {
		tmp = t_2;
	} else if (t_3 <= -4e-137) {
		tmp = t_1;
	} else if (t_3 <= 5e-58) {
		tmp = t_2;
	} else if (t_3 <= 5e+15) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	t_1 = (b / c) / z
	t_2 = (-4.0 * a) * (t / c)
	t_3 = (x * 9.0) * y
	t_4 = (9.0 * x) * (y / (c * z))
	tmp = 0
	if t_3 <= -5e+104:
		tmp = t_4
	elif t_3 <= -4e+34:
		tmp = t_2
	elif t_3 <= -4e-137:
		tmp = t_1
	elif t_3 <= 5e-58:
		tmp = t_2
	elif t_3 <= 5e+15:
		tmp = t_1
	else:
		tmp = t_4
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(b / c) / z)
	t_2 = Float64(Float64(-4.0 * a) * Float64(t / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(9.0 * x) * Float64(y / Float64(c * z)))
	tmp = 0.0
	if (t_3 <= -5e+104)
		tmp = t_4;
	elseif (t_3 <= -4e+34)
		tmp = t_2;
	elseif (t_3 <= -4e-137)
		tmp = t_1;
	elseif (t_3 <= 5e-58)
		tmp = t_2;
	elseif (t_3 <= 5e+15)
		tmp = t_1;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (b / c) / z;
	t_2 = (-4.0 * a) * (t / c);
	t_3 = (x * 9.0) * y;
	t_4 = (9.0 * x) * (y / (c * z));
	tmp = 0.0;
	if (t_3 <= -5e+104)
		tmp = t_4;
	elseif (t_3 <= -4e+34)
		tmp = t_2;
	elseif (t_3 <= -4e-137)
		tmp = t_1;
	elseif (t_3 <= 5e-58)
		tmp = t_2;
	elseif (t_3 <= 5e+15)
		tmp = t_1;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(9.0 * x), $MachinePrecision] * N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+104], t$95$4, If[LessEqual[t$95$3, -4e+34], t$95$2, If[LessEqual[t$95$3, -4e-137], t$95$1, If[LessEqual[t$95$3, 5e-58], t$95$2, If[LessEqual[t$95$3, 5e+15], t$95$1, t$95$4]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104 or 5e15 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.2

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   10. Applied rewrites 73.3%

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

   12. Applied rewrites 69.6%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999978e34 or -3.99999999999999991e-137 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999977e-58

   1. Initial program 83.5%

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

   3. Taylor expanded in z around inf

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

      1. *-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. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 54.9%

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

   if -3.99999999999999978e34 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -3.99999999999999991e-137 or 4.99999999999999977e-58 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5e15

   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 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 66.2%

      \[\leadsto \frac{\frac{b}{c}}{\color{blue}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \left(t \cdot z\right) \cdot a\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+225}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, -4, 9 \cdot \left(y \cdot x\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, t\_2 \cdot -4\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (* (* t z) a)))
   (if (<= t_1 -5e+225)
     (* (* (/ y c) 9.0) (/ x z))
     (if (<= t_1 -5e+24)
       (/ (fma t_2 -4.0 (* 9.0 (* y x))) (* z c))
       (if (<= t_1 2e+67)
         (/ (/ (fma (* (* t a) -4.0) z b) c) z)
         (if (<= t_1 5e+255)
           (/ (fma (* y x) 9.0 (* t_2 -4.0)) (* z c))
           (* (* x (/ 9.0 c)) (/ y z))))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (t * z) * a;
	double tmp;
	if (t_1 <= -5e+225) {
		tmp = ((y / c) * 9.0) * (x / z);
	} else if (t_1 <= -5e+24) {
		tmp = fma(t_2, -4.0, (9.0 * (y * x))) / (z * c);
	} else if (t_1 <= 2e+67) {
		tmp = (fma(((t * a) * -4.0), z, b) / c) / z;
	} else if (t_1 <= 5e+255) {
		tmp = fma((y * x), 9.0, (t_2 * -4.0)) / (z * c);
	} else {
		tmp = (x * (9.0 / c)) * (y / z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(t * z) * a)
	tmp = 0.0
	if (t_1 <= -5e+225)
		tmp = Float64(Float64(Float64(y / c) * 9.0) * Float64(x / z));
	elseif (t_1 <= -5e+24)
		tmp = Float64(fma(t_2, -4.0, Float64(9.0 * Float64(y * x))) / Float64(z * c));
	elseif (t_1 <= 2e+67)
		tmp = Float64(Float64(fma(Float64(Float64(t * a) * -4.0), z, b) / c) / z);
	elseif (t_1 <= 5e+255)
		tmp = Float64(fma(Float64(y * x), 9.0, Float64(t_2 * -4.0)) / Float64(z * c));
	else
		tmp = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+225], N[(N[(N[(y / c), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+24], N[(N[(t$95$2 * -4.0 + N[(9.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+67], N[(N[(N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + N[(t$95$2 * -4.0), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999981e225

   1. Initial program 78.0%

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

   3. Taylor expanded in x around 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 87.8

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

   5. Applied rewrites 87.8%

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

   if -4.99999999999999981e225 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000045e24

   1. Initial program 88.2%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 36.7%

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

   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

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 82.1

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

   11. Applied rewrites 82.1%

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

   if -5.00000000000000045e24 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999997e67

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. 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 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 77.4%

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

   if 1.99999999999999997e67 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e255

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 5.0000000000000002e255 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 44.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.2

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

   10. Applied rewrites 91.2%

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

   12. Applied rewrites 91.3%

       \[\leadsto \left(x \cdot \frac{9}{c}\right) \cdot \frac{\color{blue}{y}}{z} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y \cdot x, 9, \left(\left(t \cdot z\right) \cdot a\right) \cdot -4\right)}{z \cdot c}\\ t_2 := \left(x \cdot 9\right) \cdot y\\ t_3 := \left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{c}}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 (* (* (* t z) a) -4.0)) (* z c)))
        (t_2 (* (* x 9.0) y))
        (t_3 (* (* x (/ 9.0 c)) (/ y z))))
   (if (<= t_2 -5e+104)
     t_3
     (if (<= t_2 -5e+24)
       t_1
       (if (<= t_2 2e+67)
         (/ (/ (fma (* (* t a) -4.0) z b) c) z)
         (if (<= t_2 5e+255) t_1 t_3))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((y * x), 9.0, (((t * z) * a) * -4.0)) / (z * c);
	double t_2 = (x * 9.0) * y;
	double t_3 = (x * (9.0 / c)) * (y / z);
	double tmp;
	if (t_2 <= -5e+104) {
		tmp = t_3;
	} else if (t_2 <= -5e+24) {
		tmp = t_1;
	} else if (t_2 <= 2e+67) {
		tmp = (fma(((t * a) * -4.0), z, b) / c) / z;
	} else if (t_2 <= 5e+255) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(Float64(y * x), 9.0, Float64(Float64(Float64(t * z) * a) * -4.0)) / Float64(z * c))
	t_2 = Float64(Float64(x * 9.0) * y)
	t_3 = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z))
	tmp = 0.0
	if (t_2 <= -5e+104)
		tmp = t_3;
	elseif (t_2 <= -5e+24)
		tmp = t_1;
	elseif (t_2 <= 2e+67)
		tmp = Float64(Float64(fma(Float64(Float64(t * a) * -4.0), z, b) / c) / z);
	elseif (t_2 <= 5e+255)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(y * x), $MachinePrecision] * 9.0 + N[(N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+104], t$95$3, If[LessEqual[t$95$2, -5e+24], t$95$1, If[LessEqual[t$95$2, 2e+67], N[(N[(N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 5e+255], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104 or 5.0000000000000002e255 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 86.9

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

   10. Applied rewrites 86.9%

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

   12. Applied rewrites 86.9%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000045e24 or 1.99999999999999997e67 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e255

   1. Initial program 89.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. 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 51.4

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

   5. Applied rewrites 51.4%

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

   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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   if -5.00000000000000045e24 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999997e67

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. 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 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 77.4%

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

Alternative 8: 85.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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)) < +inf.0

   1. Initial program 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 85.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 54.0%

      \[\leadsto \left(-4 \cdot a\right) \cdot \color{blue}{\frac{t}{c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9 \cdot \frac{x}{c}, \frac{y}{z}, \mathsf{fma}\left(\frac{t \cdot a}{c}, -4, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 (<= (* (* x 9.0) y) 1e+255)
   (/ (fma (* 4.0 t) a (/ (fma (* 9.0 x) y b) (- z))) (- c))
   (fma (* 9.0 (/ x c)) (/ y z) (fma (/ (* t a) c) -4.0 (/ b (* z c))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * 9.0) * y) <= 1e+255) {
		tmp = fma((4.0 * t), a, (fma((9.0 * x), y, b) / -z)) / -c;
	} else {
		tmp = fma((9.0 * (x / c)), (y / z), fma(((t * a) / c), -4.0, (b / (z * c))));
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(x * 9.0) * y) <= 1e+255)
		tmp = Float64(fma(Float64(4.0 * t), a, Float64(fma(Float64(9.0 * x), y, b) / Float64(-z))) / Float64(-c));
	else
		tmp = fma(Float64(9.0 * Float64(x / c)), Float64(y / z), fma(Float64(Float64(t * a) / c), -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.
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[N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision], 1e+255], N[(N[(N[(4.0 * t), $MachinePrecision] * a + N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(9.0 * N[(x / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(t * a), $MachinePrecision] / c), $MachinePrecision] * -4.0 + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999988e254

   1. Initial program 81.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 91.1

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

   7. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 91.2%

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

   if 9.99999999999999988e254 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 46.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 10.8

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 95.6

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

   8. Applied rewrites 95.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9 \cdot \frac{x}{c}, \frac{y}{z}, \mathsf{fma}\left(\frac{t \cdot a}{c}, -4, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (* (* x (/ 9.0 c)) (/ y z))))
   (if (<= t_1 -5e+104)
     t_2
     (if (<= t_1 2e-25)
       (/ (fma -4.0 (* (* t z) a) b) (* z c))
       (if (<= t_1 5e+255) (/ (fma (* y x) 9.0 b) (* z c)) t_2)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (x * (9.0 / c)) * (y / z);
	double tmp;
	if (t_1 <= -5e+104) {
		tmp = t_2;
	} else if (t_1 <= 2e-25) {
		tmp = fma(-4.0, ((t * z) * a), b) / (z * c);
	} else if (t_1 <= 5e+255) {
		tmp = fma((y * x), 9.0, b) / (z * c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z))
	tmp = 0.0
	if (t_1 <= -5e+104)
		tmp = t_2;
	elseif (t_1 <= 2e-25)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(z * c));
	elseif (t_1 <= 5e+255)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+104], t$95$2, If[LessEqual[t$95$1, 2e-25], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104 or 5.0000000000000002e255 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 86.9

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

   10. Applied rewrites 86.9%

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

   12. Applied rewrites 86.9%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.00000000000000008e-25

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   if 2.00000000000000008e-25 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e255

   1. Initial program 81.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 63.9

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

   5. Applied rewrites 63.9%

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

Alternative 11: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+42} \lor \neg \left(z \leq 1.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, -\mathsf{fma}\left(\frac{x}{z} \cdot 9, y, \frac{b}{z}\right)\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 (or (<= z -8e+42) (not (<= z 1.2e-121)))
   (/ (fma (* 4.0 t) a (- (fma (* (/ x z) 9.0) y (/ b z)))) (- c))
   (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) (* z c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z <= -8e+42) || !(z <= 1.2e-121)) {
		tmp = fma((4.0 * t), a, -fma(((x / z) * 9.0), y, (b / z))) / -c;
	} else {
		tmp = fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / (z * c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((z <= -8e+42) || !(z <= 1.2e-121))
		tmp = Float64(fma(Float64(4.0 * t), a, Float64(-fma(Float64(Float64(x / z) * 9.0), y, Float64(b / z)))) / Float64(-c));
	else
		tmp = Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, 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.
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[Or[LessEqual[z, -8e+42], N[Not[LessEqual[z, 1.2e-121]], $MachinePrecision]], N[(N[(N[(4.0 * t), $MachinePrecision] * a + (-N[(N[(N[(x / z), $MachinePrecision] * 9.0), $MachinePrecision] * y + N[(b / z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000036e42 or 1.20000000000000002e-121 < z

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.1

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

   7. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 87.2%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 92.0%

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

   if -8.00000000000000036e42 < z < 1.20000000000000002e-121

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 94.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+42} \lor \neg \left(z \leq 1.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, -\mathsf{fma}\left(\frac{x}{z} \cdot 9, y, \frac{b}{z}\right)\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+104} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{c}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (or (<= t_1 -5e+104) (not (<= t_1 2e+248)))
     (* (* x (/ 9.0 c)) (/ y z))
     (/ (/ (fma (* (* t a) -4.0) z b) c) z))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if ((t_1 <= -5e+104) || !(t_1 <= 2e+248)) {
		tmp = (x * (9.0 / c)) * (y / z);
	} else {
		tmp = (fma(((t * a) * -4.0), z, b) / c) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if ((t_1 <= -5e+104) || !(t_1 <= 2e+248))
		tmp = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z));
	else
		tmp = Float64(Float64(fma(Float64(Float64(t * a) * -4.0), z, b) / c) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+104], N[Not[LessEqual[t$95$1, 2e+248]], $MachinePrecision]], N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104 or 2.00000000000000009e248 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.4

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

   10. Applied rewrites 87.4%

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

   12. Applied rewrites 87.4%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.00000000000000009e248

   1. Initial program 82.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. 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 67.8

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

   5. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 73.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -5 \cdot 10^{+104} \lor \neg \left(\left(x \cdot 9\right) \cdot y \leq 2 \cdot 10^{+248}\right):\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+104}:\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+71}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (if (<= t_1 -5e+104)
     (* (* x (/ 9.0 c)) (/ y z))
     (if (<= t_1 1e+71)
       (/ (/ (fma -4.0 (* (* t z) a) b) z) c)
       (* (* (/ y c) 9.0) (/ x z))))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+104) {
		tmp = (x * (9.0 / c)) * (y / z);
	} else if (t_1 <= 1e+71) {
		tmp = (fma(-4.0, ((t * z) * a), b) / z) / c;
	} else {
		tmp = ((y / c) * 9.0) * (x / z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+104)
		tmp = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z));
	elseif (t_1 <= 1e+71)
		tmp = Float64(Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / z) / c);
	else
		tmp = Float64(Float64(Float64(y / c) * 9.0) * Float64(x / z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+104], N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+71], N[(N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(y / c), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.9999999999999997e104

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 86.8

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.6

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

   10. Applied rewrites 84.6%

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

   12. Applied rewrites 84.6%

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

   if -4.9999999999999997e104 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1e71

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if 1e71 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 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 70.7

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

   5. Applied rewrites 70.7%

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

Alternative 14: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+42} \lor \neg \left(z \leq 1.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot 9, y, \mathsf{fma}\left(t \cdot a, -4, \frac{b}{z}\right)\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 (or (<= z -8e+42) (not (<= z 1.2e-121)))
   (/ (fma (* (/ x z) 9.0) y (fma (* t a) -4.0 (/ b z))) c)
   (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) (* z c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z <= -8e+42) || !(z <= 1.2e-121)) {
		tmp = fma(((x / z) * 9.0), y, fma((t * a), -4.0, (b / z))) / c;
	} else {
		tmp = fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / (z * c);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((z <= -8e+42) || !(z <= 1.2e-121))
		tmp = Float64(fma(Float64(Float64(x / z) * 9.0), y, fma(Float64(t * a), -4.0, Float64(b / z))) / c);
	else
		tmp = Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, 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.
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[Or[LessEqual[z, -8e+42], N[Not[LessEqual[z, 1.2e-121]], $MachinePrecision]], N[(N[(N[(N[(x / z), $MachinePrecision] * 9.0), $MachinePrecision] * y + N[(N[(t * a), $MachinePrecision] * -4.0 + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000036e42 or 1.20000000000000002e-121 < z

   1. Initial program 68.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.1

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

   7. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 87.2%

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

   10. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r* N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

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

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-/.f64 91.9

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

   12. Applied rewrites 91.9%

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

   if -8.00000000000000036e42 < z < 1.20000000000000002e-121

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \color{blue}{\left(a \cdot t\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a\right) \cdot t} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a}, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{4 \cdot z}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot z\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot z\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{-4} \cdot z\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

   4. Applied rewrites 94.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+42} \lor \neg \left(z \leq 1.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot 9, y, \mathsf{fma}\left(t \cdot a, -4, \frac{b}{z}\right)\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* (* x 9.0) y) 1e+255)
   (/ (fma (* 4.0 t) a (/ (fma (* 9.0 x) y b) (- z))) (- c))
   (* (* x (/ 9.0 c)) (/ y z))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * 9.0) * y) <= 1e+255) {
		tmp = fma((4.0 * t), a, (fma((9.0 * x), y, b) / -z)) / -c;
	} else {
		tmp = (x * (9.0 / c)) * (y / z);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(x * 9.0) * y) <= 1e+255)
		tmp = Float64(fma(Float64(4.0 * t), a, Float64(fma(Float64(9.0 * x), y, b) / Float64(-z))) / Float64(-c));
	else
		tmp = Float64(Float64(x * Float64(9.0 / c)) * Float64(y / z));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision], 1e+255], N[(N[(N[(4.0 * t), $MachinePrecision] * a + N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(x * N[(9.0 / c), $MachinePrecision]), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999988e254

   1. Initial program 81.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{\mathsf{neg}\left(z \cdot c\right)}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}}{\mathsf{neg}\left(z \cdot c\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{z \cdot c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{c \cdot z}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\mathsf{neg}\left(z\right)}} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

   4. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{-\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(4 \cdot \left(a \cdot t\right) + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{c} \cdot \left(\color{blue}{\left(4 \cdot a\right) \cdot t} + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(\color{blue}{4 \cdot a}, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{\mathsf{neg}\left(z\right)}}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{-1 \cdot z}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{-1 \cdot z}}\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{-1 \cdot z}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{-1 \cdot z}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{-1 \cdot z}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      14. mul-1-neg N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{\mathsf{neg}\left(z\right)}}\right) \]

      15. lower-neg.f64 91.1

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{-z}}\right) \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)}{c}} \]

   9. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{c}} \]

   if 9.99999999999999988e254 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 46.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{\mathsf{neg}\left(z \cdot c\right)}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}}{\mathsf{neg}\left(z \cdot c\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{z \cdot c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{c \cdot z}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\mathsf{neg}\left(z\right)}} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{-\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(4 \cdot \left(a \cdot t\right) + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{c} \cdot \left(\color{blue}{\left(4 \cdot a\right) \cdot t} + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(\color{blue}{4 \cdot a}, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{\mathsf{neg}\left(z\right)}}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{-1 \cdot z}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{-1 \cdot z}}\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{-1 \cdot z}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{-1 \cdot z}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{-1 \cdot z}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      14. mul-1-neg N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{\mathsf{neg}\left(z\right)}}\right) \]

      15. lower-neg.f64 56.2

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{-z}}\right) \]

   7. Applied rewrites 56.2%

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{c \cdot z} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot x}{c} \cdot \frac{y}{z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot x}{c} \cdot \frac{y}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot x}{c}} \cdot \frac{y}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{9 \cdot x}}{c} \cdot \frac{y}{z} \]

      7. lower-/.f64 91.6

         \[\leadsto \frac{9 \cdot x}{c} \cdot \color{blue}{\frac{y}{z}} \]

   10. Applied rewrites 91.6%

       \[\leadsto \color{blue}{\frac{9 \cdot x}{c} \cdot \frac{y}{z}} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.6%

       \[\leadsto \left(x \cdot \frac{9}{c}\right) \cdot \frac{\color{blue}{y}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{c}\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+217}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, -9 \cdot \frac{y \cdot x}{z}\right)}{-c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 -1.45e+217)
   (* (* -4.0 a) (/ t c))
   (if (<= z 1.7e+108)
     (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) (* z c))
     (/ (fma (* 4.0 t) a (* -9.0 (/ (* y x) z))) (- c)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (z <= -1.45e+217) {
		tmp = (-4.0 * a) * (t / c);
	} else if (z <= 1.7e+108) {
		tmp = fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / (z * c);
	} else {
		tmp = fma((4.0 * t), a, (-9.0 * ((y * x) / z))) / -c;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (z <= -1.45e+217)
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	elseif (z <= 1.7e+108)
		tmp = Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, b)) / Float64(z * c));
	else
		tmp = Float64(fma(Float64(4.0 * t), a, Float64(-9.0 * Float64(Float64(y * x) / z))) / Float64(-c));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[z, -1.45e+217], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+108], N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * t), $MachinePrecision] * a + N[(-9.0 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999992e217

   1. Initial program 35.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 49.4

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.1%

      \[\leadsto \left(-4 \cdot a\right) \cdot \color{blue}{\frac{t}{c}} \]

   if -1.44999999999999992e217 < z < 1.69999999999999998e108

   1. Initial program 87.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)\right)} + b}{z \cdot c} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \color{blue}{\left(a \cdot t\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a\right) \cdot t} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot a}, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{z \cdot 4}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{4 \cdot z}\right)\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot z\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot z\right)} \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{-4} \cdot z\right) \cdot a, t, \left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

   4. Applied rewrites 89.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}}{z \cdot c} \]

   if 1.69999999999999998e108 < z

   1. Initial program 56.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{\mathsf{neg}\left(z \cdot c\right)}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}}{\mathsf{neg}\left(z \cdot c\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{z \cdot c}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\mathsf{neg}\left(\color{blue}{c \cdot z}\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{-1 \cdot \left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)}{\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\mathsf{neg}\left(z\right)}} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \left(\mathsf{neg}\left(\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\right)\right)}{z}} \]

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{-1}{c} \cdot \frac{-\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(-1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z} + 4 \cdot \left(a \cdot t\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\left(4 \cdot \left(a \cdot t\right) + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{c} \cdot \left(\color{blue}{\left(4 \cdot a\right) \cdot t} + -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(\color{blue}{4 \cdot a}, t, -1 \cdot \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\mathsf{neg}\left(\frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{\mathsf{neg}\left(z\right)}}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{-1 \cdot z}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{-1 \cdot z}}\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{-1 \cdot z}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{-1 \cdot z}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{-1 \cdot z}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{-1 \cdot z}\right) \]

      14. mul-1-neg N/A

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{\mathsf{neg}\left(z\right)}}\right) \]

      15. lower-neg.f64 89.6

          \[\leadsto \frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{\color{blue}{-z}}\right) \]

   7. Applied rewrites 89.6%

      \[\leadsto \frac{-1}{c} \cdot \color{blue}{\mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{c}} \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \mathsf{fma}\left(4 \cdot a, t, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{-z}\right)}{c}} \]

   9. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(4 \cdot t, a, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{-z}\right)}{c}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{-\left(-9 \cdot \frac{x \cdot y}{z} + \color{blue}{4 \cdot \left(a \cdot t\right)}\right)}{c} \]
   11. Step-by-step derivation

   12. Applied rewrites 80.1%

       \[\leadsto \frac{-\mathsf{fma}\left(4 \cdot t, \color{blue}{a}, -9 \cdot \frac{y \cdot x}{z}\right)}{c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+217}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4 \cdot t, a, -9 \cdot \frac{y \cdot x}{z}\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10} \lor \neg \left(a \leq 2.4 \cdot 10^{+175}\right):\\ \;\;\;\;\left(t \cdot \frac{a}{c}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= a -1.6e-10) (not (<= a 2.4e+175)))
   (* (* t (/ a c)) -4.0)
   (/ (/ (fma (* y x) 9.0 b) c) z)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a <= -1.6e-10) || !(a <= 2.4e+175)) {
		tmp = (t * (a / c)) * -4.0;
	} else {
		tmp = (fma((y * x), 9.0, b) / c) / z;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((a <= -1.6e-10) || !(a <= 2.4e+175))
		tmp = Float64(Float64(t * Float64(a / c)) * -4.0);
	else
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c) / z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[a, -1.6e-10], N[Not[LessEqual[a, 2.4e+175]], $MachinePrecision]], N[(N[(t * N[(a / c), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5999999999999999e-10 or 2.4e175 < a

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 57.3

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.5%

      \[\leadsto \left(t \cdot \frac{a}{c}\right) \cdot -4 \]

   if -1.5999999999999999e-10 < a < 2.4e175

   1. Initial program 81.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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 75.0

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{c}}{z} \]

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10} \lor \neg \left(a \leq 2.4 \cdot 10^{+175}\right):\\ \;\;\;\;\left(t \cdot \frac{a}{c}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10} \lor \neg \left(a \leq 2.5 \cdot 10^{+175}\right):\\ \;\;\;\;\left(t \cdot \frac{a}{c}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
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 (or (<= a -1.6e-10) (not (<= a 2.5e+175)))
   (* (* t (/ a c)) -4.0)
   (/ (fma (* y x) 9.0 b) (* z c))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((a <= -1.6e-10) || !(a <= 2.5e+175)) {
		tmp = (t * (a / c)) * -4.0;
	} 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])
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 <= -1.6e-10) || !(a <= 2.5e+175))
		tmp = Float64(Float64(t * Float64(a / c)) * -4.0);
	else
		tmp = Float64(fma(Float64(y * x), 9.0, 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.
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[Or[LessEqual[a, -1.6e-10], N[Not[LessEqual[a, 2.5e+175]], $MachinePrecision]], N[(N[(t * N[(a / c), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5999999999999999e-10 or 2.5e175 < a

   1. Initial program 74.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 57.3

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.5%

      \[\leadsto \left(t \cdot \frac{a}{c}\right) \cdot -4 \]

   if -1.5999999999999999e-10 < a < 2.5e175

   1. Initial program 81.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. 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 73.5

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z \cdot c} \]

   5. Applied rewrites 73.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot x, 9, b\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-10} \lor \neg \left(a \leq 2.5 \cdot 10^{+175}\right):\\ \;\;\;\;\left(t \cdot \frac{a}{c}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+135}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -4.8e+135)
   (* (/ (* a t) c) -4.0)
   (if (<= t 1.4e-84) (/ (/ b c) z) (* (* -4.0 a) (/ t c)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -4.8e+135) {
		tmp = ((a * t) / c) * -4.0;
	} else if (t <= 1.4e-84) {
		tmp = (b / c) / z;
	} else {
		tmp = (-4.0 * a) * (t / c);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (t <= (-4.8d+135)) then
        tmp = ((a * t) / c) * (-4.0d0)
    else if (t <= 1.4d-84) then
        tmp = (b / c) / z
    else
        tmp = ((-4.0d0) * a) * (t / c)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -4.8e+135) {
		tmp = ((a * t) / c) * -4.0;
	} else if (t <= 1.4e-84) {
		tmp = (b / c) / z;
	} else {
		tmp = (-4.0 * a) * (t / c);
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if t <= -4.8e+135:
		tmp = ((a * t) / c) * -4.0
	elif t <= 1.4e-84:
		tmp = (b / c) / z
	else:
		tmp = (-4.0 * a) * (t / c)
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -4.8e+135)
		tmp = Float64(Float64(Float64(a * t) / c) * -4.0);
	elseif (t <= 1.4e-84)
		tmp = Float64(Float64(b / c) / z);
	else
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (t <= -4.8e+135)
		tmp = ((a * t) / c) * -4.0;
	elseif (t <= 1.4e-84)
		tmp = (b / c) / z;
	else
		tmp = (-4.0 * a) * (t / c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -4.8e+135], N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t, 1.4e-84], N[(N[(b / c), $MachinePrecision] / z), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.79999999999999995e135

   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 z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 63.3

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

   if -4.79999999999999995e135 < t < 1.39999999999999991e-84

   1. Initial program 79.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 36.2

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

      \[\leadsto \frac{\frac{b}{c}}{\color{blue}{z}} \]

   if 1.39999999999999991e-84 < t

   1. Initial program 77.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. *-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. lower-*.f64 43.4

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.1%

      \[\leadsto \left(-4 \cdot a\right) \cdot \color{blue}{\frac{t}{c}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 50.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-43} \lor \neg \left(t \leq 8.8 \cdot 10^{-153}\right):\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= t -1.6e-43) (not (<= t 8.8e-153)))
   (* (* -4.0 a) (/ t c))
   (/ b (* c z))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t <= -1.6e-43) || !(t <= 8.8e-153)) {
		tmp = (-4.0 * a) * (t / c);
	} else {
		tmp = b / (c * z);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((t <= (-1.6d-43)) .or. (.not. (t <= 8.8d-153))) then
        tmp = ((-4.0d0) * a) * (t / c)
    else
        tmp = b / (c * z)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((t <= -1.6e-43) || !(t <= 8.8e-153)) {
		tmp = (-4.0 * a) * (t / c);
	} else {
		tmp = b / (c * z);
	}
	return tmp;
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	tmp = 0
	if (t <= -1.6e-43) or not (t <= 8.8e-153):
		tmp = (-4.0 * a) * (t / c)
	else:
		tmp = b / (c * z)
	return tmp

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((t <= -1.6e-43) || !(t <= 8.8e-153))
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c));
	else
		tmp = Float64(b / Float64(c * z));
	end
	return tmp
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((t <= -1.6e-43) || ~((t <= 8.8e-153)))
		tmp = (-4.0 * a) * (t / c);
	else
		tmp = b / (c * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[t, -1.6e-43], N[Not[LessEqual[t, 8.8e-153]], $MachinePrecision]], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c), $MachinePrecision]), $MachinePrecision], N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.59999999999999992e-43 or 8.80000000000000003e-153 < t

   1. Initial program 78.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. 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. lower-*.f64 46.6

         \[\leadsto \frac{\color{blue}{a \cdot t}}{c} \cdot -4 \]

   5. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.6%

      \[\leadsto \left(-4 \cdot a\right) \cdot \color{blue}{\frac{t}{c}} \]

   if -1.59999999999999992e-43 < t < 8.80000000000000003e-153

   1. Initial program 78.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 39.0

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-43} \lor \neg \left(t \leq 8.8 \cdot 10^{-153}\right):\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\\\ [x, y, z, t, a, b, c] = \mathsf{sort}([x, y, z, t, a, b, c])\\ \\ \frac{b}{c \cdot z} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c) :precision binary64 (/ b (* c z)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c);
assert(x < y && y < z && z < t && t < a && a < b && b < c);
double code(double x, double y, double z, double t, double a, double b, double c) {
	return b / (c * z);
}

Fortran

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / (c * z)
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c;
assert x < y && y < z && z < t && t < a && a < b && b < c;
public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return b / (c * z);
}

Python

[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
[x, y, z, t, a, b, c] = sort([x, y, z, t, a, b, c])
def code(x, y, z, t, a, b, c):
	return b / (c * z)

Julia

x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
x, y, z, t, a, b, c = sort([x, y, z, t, a, b, c])
function code(x, y, z, t, a, b, c)
	return Float64(b / Float64(c * z))
end

MATLAB

x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
x, y, z, t, a, b, c = num2cell(sort([x, y, z, t, a, b, c])){:}
function tmp = code(x, y, z, t, a, b, c)
	tmp = b / (c * z);
end

Wolfram

NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_] := N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
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 31.1

      \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

5. Applied rewrites 31.1%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 80.9% 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 2024307 
(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)))