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

Percentage Accurate: 79.3% → 86.6%

Time: 16.2s

Alternatives: 19

Speedup: 0.3×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.3× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_1 0.0)
      (/ (/ (fma (* z (* t a)) -4.0 (fma x (* 9.0 y) b)) z) c_m)
      (if (<= t_1 INFINITY)
        (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c_m))
        (* a (/ (* t -4.0) c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (fma((z * (t * a)), -4.0, fma(x, (9.0 * y), b)) / z) / c_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c_m);
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(z * Float64(t * a)), -4.0, fma(x, Float64(9.0 * y), b)) / z) / c_m);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c_m));
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(z * N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 89.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 91.1%

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

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

   1. Initial program 88.1%

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

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.1

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

   4. Applied rewrites 88.1%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 89.1%

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

Alternative 2: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c\_m}, \frac{b}{c\_m}\right)}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c_m)))
        (t_2 (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c_m))))
   (*
    c_s
    (if (<= t_1 -5e-284)
      t_2
      (if (<= t_1 0.0)
        (/ (fma 9.0 (/ (* x y) c_m) (/ b c_m)) z)
        (if (<= t_1 INFINITY) t_2 (* a (/ (* t -4.0) c_m))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double t_2 = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c_m);
	double tmp;
	if (t_1 <= -5e-284) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(9.0, ((x * y) / c_m), (b / c_m)) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	t_2 = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= -5e-284)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(9.0, Float64(Float64(x * y) / c_m), Float64(b / c_m)) / z);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e-284], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(9.0 * N[(N[(x * y), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(b / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -4.99999999999999973e-284 or -0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 88.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if -4.99999999999999973e-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 39.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{b}{c}\right)}{z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c_m)))
        (t_2 (/ (fma (* x 9.0) y (fma a (* -4.0 (* z t)) b)) (* z c_m))))
   (*
    c_s
    (if (<= t_1 -2e-216)
      t_2
      (if (<= t_1 0.0)
        (/ (/ (fma z (* (* t a) -4.0) b) z) c_m)
        (if (<= t_1 INFINITY) t_2 (* a (/ (* t -4.0) c_m))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double t_2 = fma((x * 9.0), y, fma(a, (-4.0 * (z * t)), b)) / (z * c_m);
	double tmp;
	if (t_1 <= -2e-216) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma(z, ((t * a) * -4.0), b) / z) / c_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	t_2 = Float64(fma(Float64(x * 9.0), y, fma(a, Float64(-4.0 * Float64(z * t)), b)) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= -2e-216)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / z) / c_m);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := 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$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-216], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -2.0000000000000001e-216 or -0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 88.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if -2.0000000000000001e-216 < (/.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 41.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.3× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma a (* -4.0 (* z t)) b))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 0.0)
      (/ (/ (fma x (* 9.0 y) t_1) z) c_m)
      (if (<= t_2 INFINITY)
        (/ (fma (* x 9.0) y t_1) (* z c_m))
        (* a (/ (* t -4.0) c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(a, (-4.0 * (z * t)), b);
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (fma(x, (9.0 * y), t_1) / z) / c_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, t_1) / (z * c_m);
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(a, Float64(-4.0 * Float64(z * t)), b)
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_1) / z) / c_m);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, t_1) / Float64(z * c_m));
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$2 = 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$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, 0.0], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$1), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$1), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 89.1%

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

   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 88.1%

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

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.1

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

   4. Applied rewrites 88.1%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 87.9%

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

Alternative 5: 86.9% accurate, 0.3× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma a (* -4.0 (* z t)) b))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c_m))))
   (*
    c_s
    (if (<= t_2 0.0)
      (/ (/ (fma x (* 9.0 y) t_1) c_m) z)
      (if (<= t_2 INFINITY)
        (/ (fma (* x 9.0) y t_1) (* z c_m))
        (* a (/ (* t -4.0) c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(a, (-4.0 * (z * t)), b);
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c_m);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (fma(x, (9.0 * y), t_1) / c_m) / z;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((x * 9.0), y, t_1) / (z * c_m);
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(a, Float64(-4.0 * Float64(z * t)), b)
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_1) / c_m) / z);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(x * 9.0), y, t_1) / Float64(z * c_m));
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$2 = 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$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, 0.0], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$1), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$1), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 79.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 87.1%

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

   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 88.1%

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

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.1

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

   4. Applied rewrites 88.1%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 14.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 86.8%

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

Alternative 6: 53.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := 9 \cdot \left(x \cdot \frac{y}{z \cdot c\_m}\right)\\ t_2 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* 9.0 (* x (/ y (* z c_m))))) (t_2 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_2 -5e+71)
      t_1
      (if (<= t_2 -5e-260)
        (/ b (* z c_m))
        (if (<= t_2 200.0) (* a (/ (* t -4.0) c_m)) t_1))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (z * c_m)));
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -5e+71) {
		tmp = t_1;
	} else if (t_2 <= -5e-260) {
		tmp = b / (z * c_m);
	} else if (t_2 <= 200.0) {
		tmp = a * ((t * -4.0) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 9.0d0 * (x * (y / (z * c_m)))
    t_2 = (x * 9.0d0) * y
    if (t_2 <= (-5d+71)) then
        tmp = t_1
    else if (t_2 <= (-5d-260)) then
        tmp = b / (z * c_m)
    else if (t_2 <= 200.0d0) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (z * c_m)));
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -5e+71) {
		tmp = t_1;
	} else if (t_2 <= -5e-260) {
		tmp = b / (z * c_m);
	} else if (t_2 <= 200.0) {
		tmp = a * ((t * -4.0) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = 9.0 * (x * (y / (z * c_m)))
	t_2 = (x * 9.0) * y
	tmp = 0
	if t_2 <= -5e+71:
		tmp = t_1
	elif t_2 <= -5e-260:
		tmp = b / (z * c_m)
	elif t_2 <= 200.0:
		tmp = a * ((t * -4.0) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(9.0 * Float64(x * Float64(y / Float64(z * c_m))))
	t_2 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_2 <= -5e+71)
		tmp = t_1;
	elseif (t_2 <= -5e-260)
		tmp = Float64(b / Float64(z * c_m));
	elseif (t_2 <= 200.0)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = 9.0 * (x * (y / (z * c_m)));
	t_2 = (x * 9.0) * y;
	tmp = 0.0;
	if (t_2 <= -5e+71)
		tmp = t_1;
	elseif (t_2 <= -5e-260)
		tmp = b / (z * c_m);
	elseif (t_2 <= 200.0)
		tmp = a * ((t * -4.0) / c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(9.0 * N[(x * N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+71], t$95$1, If[LessEqual[t$95$2, -5e-260], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999972e71 or 200 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

   if -4.99999999999999972e71 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000003e-260

   1. Initial program 84.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   if -5.0000000000000003e-260 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 200

   1. Initial program 79.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 70.1

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

   7. Applied rewrites 70.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -5 \cdot 10^{+71}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 200:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\frac{z \cdot 0.1111111111111111}{y} \cdot \frac{c\_m}{x}}\\ \mathbf{elif}\;t\_1 \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \left(z \cdot \left(t \cdot a\right)\right) \cdot -4\right)}{z \cdot c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -5e+138)
      (/ 1.0 (* (/ (* z 0.1111111111111111) y) (/ c_m x)))
      (if (<= t_1 1e+31)
        (/ (/ (fma z (* (* t a) -4.0) b) z) c_m)
        (/ (fma (* x 9.0) y (* (* z (* t a)) -4.0)) (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -5e+138) {
		tmp = 1.0 / (((z * 0.1111111111111111) / y) * (c_m / x));
	} else if (t_1 <= 1e+31) {
		tmp = (fma(z, ((t * a) * -4.0), b) / z) / c_m;
	} else {
		tmp = fma((x * 9.0), y, ((z * (t * a)) * -4.0)) / (z * c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -5e+138)
		tmp = Float64(1.0 / Float64(Float64(Float64(z * 0.1111111111111111) / y) * Float64(c_m / x)));
	elseif (t_1 <= 1e+31)
		tmp = Float64(Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / z) / c_m);
	else
		tmp = Float64(fma(Float64(x * 9.0), y, Float64(Float64(z * Float64(t * a)) * -4.0)) / Float64(z * c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+138], N[(1.0 / N[(N[(N[(z * 0.1111111111111111), $MachinePrecision] / y), $MachinePrecision] * N[(c$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+31], N[(N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(N[(z * N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000016e138

   1. Initial program 71.3%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 71.4

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+l- N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{z \cdot c}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      15. associate-+l- N/A

          \[\leadsto \frac{1}{\frac{z \cdot c}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      16. neg-sub0 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{9} \cdot \frac{c \cdot z}{x \cdot y}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{9} \cdot \left(c \cdot z\right)}{x \cdot y}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{9} \cdot \left(c \cdot z\right)}{x \cdot y}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{9} \cdot \left(c \cdot z\right)}}{x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{9} \cdot \color{blue}{\left(z \cdot c\right)}}{x \cdot y}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{9} \cdot \color{blue}{\left(z \cdot c\right)}}{x \cdot y}} \]

      6. lower-*.f64 64.2

         \[\leadsto \frac{1}{\frac{0.1111111111111111 \cdot \left(z \cdot c\right)}{\color{blue}{x \cdot y}}} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.1111111111111111 \cdot \left(z \cdot c\right)}{x \cdot y}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 81.3%

      \[\leadsto \frac{1}{\frac{z \cdot 0.1111111111111111}{y} \cdot \color{blue}{\frac{c}{x}}} \]

   if -5.00000000000000016e138 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999996e30

   1. Initial program 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if 9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.1%

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

   3. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 72.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\frac{z \cdot 0.1111111111111111}{y} \cdot \frac{c}{x}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \left(z \cdot \left(t \cdot a\right)\right) \cdot -4\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \left(z \cdot \left(t \cdot a\right)\right) \cdot -4\right)}{z \cdot c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+35)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (if (<= t_1 1e+31)
        (/ (/ (fma z (* (* t a) -4.0) b) z) c_m)
        (/ (fma (* x 9.0) y (* (* z (* t a)) -4.0)) (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else if (t_1 <= 1e+31) {
		tmp = (fma(z, ((t * a) * -4.0), b) / z) / c_m;
	} else {
		tmp = fma((x * 9.0), y, ((z * (t * a)) * -4.0)) / (z * c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	elseif (t_1 <= 1e+31)
		tmp = Float64(Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / z) / c_m);
	else
		tmp = Float64(fma(Float64(x * 9.0), y, Float64(Float64(z * Float64(t * a)) * -4.0)) / Float64(z * c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+35], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+31], N[(N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(N[(z * N[(t * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999999e35

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999996e30

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 88.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 83.5

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

   9. Applied rewrites 83.5%

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

   if 9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.1%

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

   3. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 72.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \left(z \cdot \left(t \cdot a\right)\right) \cdot -4\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+35)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (if (<= t_1 1e+31)
        (/ (/ (fma z (* (* t a) -4.0) b) z) c_m)
        (/ (fma a (* -4.0 (* z t)) (* 9.0 (* x y))) (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else if (t_1 <= 1e+31) {
		tmp = (fma(z, ((t * a) * -4.0), b) / z) / c_m;
	} else {
		tmp = fma(a, (-4.0 * (z * t)), (9.0 * (x * y))) / (z * c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	elseif (t_1 <= 1e+31)
		tmp = Float64(Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / z) / c_m);
	else
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), Float64(9.0 * Float64(x * y))) / Float64(z * c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+35], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+31], N[(N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999999e35

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999996e30

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 88.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 83.5

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

   9. Applied rewrites 83.5%

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

   if 9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.1%

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

   3. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), 9 \cdot \left(x \cdot y\right)\right)}{z \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 9\right) \cdot \frac{y}{z}}{c\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+35)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (if (<= t_1 2e+186)
        (/ (/ (fma z (* (* t a) -4.0) b) z) c_m)
        (/ (* (* x 9.0) (/ y z)) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else if (t_1 <= 2e+186) {
		tmp = (fma(z, ((t * a) * -4.0), b) / z) / c_m;
	} else {
		tmp = ((x * 9.0) * (y / z)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	elseif (t_1 <= 2e+186)
		tmp = Float64(Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / z) / c_m);
	else
		tmp = Float64(Float64(Float64(x * 9.0) * Float64(y / z)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+35], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+186], N[(N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999999e35

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999996e186

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 88.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 78.7

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

   9. Applied rewrites 78.7%

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

   if 1.99999999999999996e186 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 68.0

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

   9. Applied rewrites 68.0%

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

   11. Applied rewrites 69.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 9\right) \cdot \frac{y}{z}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+35)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (if (<= t_1 1e+137)
        (/ (fma a (* -4.0 (* z t)) b) (* z c_m))
        (* 9.0 (* x (/ y (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else if (t_1 <= 1e+137) {
		tmp = fma(a, (-4.0 * (z * t)), b) / (z * c_m);
	} else {
		tmp = 9.0 * (x * (y / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	elseif (t_1 <= 1e+137)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), b) / Float64(z * c_m));
	else
		tmp = Float64(9.0 * Float64(x * Float64(y / Float64(z * c_m))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+35], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+137], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999999e35

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1e137

   1. Initial program 81.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if 1e137 < (*.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. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -2e+35)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (if (<= t_1 2e+186)
        (/ (fma z (* (* t a) -4.0) b) (* z c_m))
        (* 9.0 (* x (/ y (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else if (t_1 <= 2e+186) {
		tmp = fma(z, ((t * a) * -4.0), b) / (z * c_m);
	} else {
		tmp = 9.0 * (x * (y / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	elseif (t_1 <= 2e+186)
		tmp = Float64(fma(z, Float64(Float64(t * a) * -4.0), b) / Float64(z * c_m));
	else
		tmp = Float64(9.0 * Float64(x * Float64(y / Float64(z * c_m))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+35], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+186], N[(N[(z * N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999999e35

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.9999999999999999e35 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999996e186

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 84.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 88.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 1.99999999999999996e186 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 2 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \left(t \cdot a\right) \cdot -4, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{\frac{c\_m}{t \cdot -4}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= t -1.55e+145)
    (/ a (/ c_m (* t -4.0)))
    (if (<= t 5e-165)
      (/ (fma 9.0 (* x y) b) (* z c_m))
      (* a (/ (* t -4.0) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -1.55e+145) {
		tmp = a / (c_m / (t * -4.0));
	} else if (t <= 5e-165) {
		tmp = fma(9.0, (x * y), b) / (z * c_m);
	} else {
		tmp = a * ((t * -4.0) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (t <= -1.55e+145)
		tmp = Float64(a / Float64(c_m / Float64(t * -4.0)));
	elseif (t <= 5e-165)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(z * c_m));
	else
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[t, -1.55e+145], N[(a / N[(c$95$m / N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-165], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.54999999999999994e145

   1. Initial program 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   11. Applied rewrites 66.7%

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

   if -1.54999999999999994e145 < t < 4.99999999999999981e-165

   1. Initial program 82.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if 4.99999999999999981e-165 < t

   1. Initial program 68.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 49.9

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{\frac{c}{t \cdot -4}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{b}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= t -6.2)
    (* a (/ (* t -4.0) c_m))
    (if (<= t 1.05e-242) (/ (/ b z) c_m) (* (* a -4.0) (/ t c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 1.05e-242) {
		tmp = (b / z) / c_m;
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if (t <= (-6.2d0)) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else if (t <= 1.05d-242) then
        tmp = (b / z) / c_m
    else
        tmp = (a * (-4.0d0)) * (t / c_m)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 1.05e-242) {
		tmp = (b / z) / c_m;
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if t <= -6.2:
		tmp = a * ((t * -4.0) / c_m)
	elif t <= 1.05e-242:
		tmp = (b / z) / c_m
	else:
		tmp = (a * -4.0) * (t / c_m)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (t <= -6.2)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	elseif (t <= 1.05e-242)
		tmp = Float64(Float64(b / z) / c_m);
	else
		tmp = Float64(Float64(a * -4.0) * Float64(t / c_m));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (t <= -6.2)
		tmp = a * ((t * -4.0) / c_m);
	elseif (t <= 1.05e-242)
		tmp = (b / z) / c_m;
	else
		tmp = (a * -4.0) * (t / c_m);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[t, -6.2], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-242], N[(N[(b / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.20000000000000018

   1. Initial program 87.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 58.3

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]

   if -6.20000000000000018 < t < 1.05000000000000009e-242

   1. Initial program 80.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]
   6. Step-by-step derivation

      1. lower-/.f64 41.1

         \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]

   7. Applied rewrites 41.1%

      \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]

   if 1.05000000000000009e-242 < t

   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. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(9 \cdot y\right) + \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}}{z}}{c} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right) + \color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + b\right)}}{z}}{c} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot \left(9 \cdot y\right) + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}}{z}}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot \color{blue}{\left(9 \cdot y\right)} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(x \cdot 9\right) \cdot y} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z}}{c} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(-4 \cdot \left(z \cdot t\right)\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(z \cdot t\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot -4\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot \left(t \cdot z\right), -4, \left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(a \cdot \color{blue}{\left(t \cdot z\right)}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      21. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      22. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

   6. Applied rewrites 80.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, -4, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z}}{c} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot \frac{t}{c}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot -4\right)} \cdot \frac{t}{c} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 46.9

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   9. Applied rewrites 46.9%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(a \cdot -4\right) \cdot \color{blue}{\frac{t}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= t -6.2)
    (* a (/ (* t -4.0) c_m))
    (if (<= t 1.05e-242) (/ (/ b c_m) z) (* (* a -4.0) (/ t c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 1.05e-242) {
		tmp = (b / c_m) / z;
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if (t <= (-6.2d0)) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else if (t <= 1.05d-242) then
        tmp = (b / c_m) / z
    else
        tmp = (a * (-4.0d0)) * (t / c_m)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 1.05e-242) {
		tmp = (b / c_m) / z;
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if t <= -6.2:
		tmp = a * ((t * -4.0) / c_m)
	elif t <= 1.05e-242:
		tmp = (b / c_m) / z
	else:
		tmp = (a * -4.0) * (t / c_m)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (t <= -6.2)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	elseif (t <= 1.05e-242)
		tmp = Float64(Float64(b / c_m) / z);
	else
		tmp = Float64(Float64(a * -4.0) * Float64(t / c_m));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (t <= -6.2)
		tmp = a * ((t * -4.0) / c_m);
	elseif (t <= 1.05e-242)
		tmp = (b / c_m) / z;
	else
		tmp = (a * -4.0) * (t / c_m);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[t, -6.2], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-242], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.20000000000000018

   1. Initial program 87.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 58.3

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]

   if -6.20000000000000018 < t < 1.05000000000000009e-242

   1. Initial program 80.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 35.0

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.6%

      \[\leadsto \frac{\frac{b}{c}}{\color{blue}{z}} \]

   if 1.05000000000000009e-242 < t

   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. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(9 \cdot y\right) + \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}}{z}}{c} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right) + \color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + b\right)}}{z}}{c} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot \left(9 \cdot y\right) + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}}{z}}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot \color{blue}{\left(9 \cdot y\right)} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(x \cdot 9\right) \cdot y} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z}}{c} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(-4 \cdot \left(z \cdot t\right)\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(z \cdot t\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot -4\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot \left(t \cdot z\right), -4, \left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(a \cdot \color{blue}{\left(t \cdot z\right)}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      21. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      22. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

   6. Applied rewrites 80.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, -4, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z}}{c} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot \frac{t}{c}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot -4\right)} \cdot \frac{t}{c} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 46.9

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   9. Applied rewrites 46.9%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(a \cdot -4\right) \cdot \color{blue}{\frac{t}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c\_m}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= t -6.2)
    (* a (/ (* t -4.0) c_m))
    (if (<= t 2.8e-243) (/ b (* z c_m)) (* (* a -4.0) (/ t c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 2.8e-243) {
		tmp = b / (z * c_m);
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if (t <= (-6.2d0)) then
        tmp = a * ((t * (-4.0d0)) / c_m)
    else if (t <= 2.8d-243) then
        tmp = b / (z * c_m)
    else
        tmp = (a * (-4.0d0)) * (t / c_m)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (t <= -6.2) {
		tmp = a * ((t * -4.0) / c_m);
	} else if (t <= 2.8e-243) {
		tmp = b / (z * c_m);
	} else {
		tmp = (a * -4.0) * (t / c_m);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if t <= -6.2:
		tmp = a * ((t * -4.0) / c_m)
	elif t <= 2.8e-243:
		tmp = b / (z * c_m)
	else:
		tmp = (a * -4.0) * (t / c_m)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (t <= -6.2)
		tmp = Float64(a * Float64(Float64(t * -4.0) / c_m));
	elseif (t <= 2.8e-243)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = Float64(Float64(a * -4.0) * Float64(t / c_m));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (t <= -6.2)
		tmp = a * ((t * -4.0) / c_m);
	elseif (t <= 2.8e-243)
		tmp = b / (z * c_m);
	else
		tmp = (a * -4.0) * (t / c_m);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[t, -6.2], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e-243], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.20000000000000018

   1. Initial program 87.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 58.3

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]

   if -6.20000000000000018 < t < 2.79999999999999994e-243

   1. Initial program 80.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 35.0

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]

   if 2.79999999999999994e-243 < t

   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. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(9 \cdot y\right) + \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}}{z}}{c} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{x \cdot \left(9 \cdot y\right) + \color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + b\right)}}{z}}{c} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot \left(9 \cdot y\right) + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}}{z}}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(x \cdot \color{blue}{\left(9 \cdot y\right)} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(x \cdot 9\right) \cdot y} + a \cdot \left(-4 \cdot \left(z \cdot t\right)\right)\right) + b}{z}}{c} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(x \cdot 9\right) \cdot y\right)} + b}{z}}{c} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(-4 \cdot \left(z \cdot t\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(-4 \cdot \left(z \cdot t\right)\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(z \cdot t\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot -4\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot \left(t \cdot z\right), -4, \left(x \cdot 9\right) \cdot y + b\right)}}{z}}{c} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(a \cdot \color{blue}{\left(t \cdot z\right)}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right) \cdot z}, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(t \cdot a\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      21. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

      22. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(a \cdot t\right)} \cdot z, -4, \left(x \cdot 9\right) \cdot y + b\right)}{z}}{c} \]

   6. Applied rewrites 80.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(a \cdot t\right) \cdot z, -4, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}}{z}}{c} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -4 \cdot \color{blue}{\left(a \cdot \frac{t}{c}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-4 \cdot a\right) \cdot \frac{t}{c}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot -4\right)} \cdot \frac{t}{c} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 46.9

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   9. Applied rewrites 46.9%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.9%

       \[\leadsto \left(a \cdot -4\right) \cdot \color{blue}{\frac{t}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := a \cdot \frac{t \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* a (/ (* t -4.0) c_m))))
   (* c_s (if (<= t -6.2) t_1 (if (<= t 2.8e-243) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = a * ((t * -4.0) / c_m);
	double tmp;
	if (t <= -6.2) {
		tmp = t_1;
	} else if (t <= 2.8e-243) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * ((t * (-4.0d0)) / c_m)
    if (t <= (-6.2d0)) then
        tmp = t_1
    else if (t <= 2.8d-243) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = a * ((t * -4.0) / c_m);
	double tmp;
	if (t <= -6.2) {
		tmp = t_1;
	} else if (t <= 2.8e-243) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = a * ((t * -4.0) / c_m)
	tmp = 0
	if t <= -6.2:
		tmp = t_1
	elif t <= 2.8e-243:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(a * Float64(Float64(t * -4.0) / c_m))
	tmp = 0.0
	if (t <= -6.2)
		tmp = t_1;
	elseif (t <= 2.8e-243)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = a * ((t * -4.0) / c_m);
	tmp = 0.0;
	if (t <= -6.2)
		tmp = t_1;
	elseif (t <= 2.8e-243)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -6.2], t$95$1, If[LessEqual[t, 2.8e-243], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.20000000000000018 or 2.79999999999999994e-243 < t

   1. Initial program 75.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-4 \cdot \frac{t}{c}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      7. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{-4 \cdot t}{c}} \]

      8. lower-*.f64 51.0

         \[\leadsto a \cdot \frac{\color{blue}{-4 \cdot t}}{c} \]

   7. Applied rewrites 51.0%

      \[\leadsto \color{blue}{a \cdot \frac{-4 \cdot t}{c}} \]

   if -6.20000000000000018 < t < 2.79999999999999994e-243

   1. Initial program 80.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 35.0

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -1.82 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 800000000000:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (* (* t a) -4.0) c_m)))
   (*
    c_s
    (if (<= a -1.82e-152)
      t_1
      (if (<= a 800000000000.0) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((t * a) * -4.0) / c_m;
	double tmp;
	if (a <= -1.82e-152) {
		tmp = t_1;
	} else if (a <= 800000000000.0) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t * a) * (-4.0d0)) / c_m
    if (a <= (-1.82d-152)) then
        tmp = t_1
    else if (a <= 800000000000.0d0) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((t * a) * -4.0) / c_m;
	double tmp;
	if (a <= -1.82e-152) {
		tmp = t_1;
	} else if (a <= 800000000000.0) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((t * a) * -4.0) / c_m
	tmp = 0
	if a <= -1.82e-152:
		tmp = t_1
	elif a <= 800000000000.0:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(t * a) * -4.0) / c_m)
	tmp = 0.0
	if (a <= -1.82e-152)
		tmp = t_1;
	elseif (a <= 800000000000.0)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((t * a) * -4.0) / c_m;
	tmp = 0.0;
	if (a <= -1.82e-152)
		tmp = t_1;
	elseif (a <= 800000000000.0)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -1.82e-152], t$95$1, If[LessEqual[a, 800000000000.0], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.82000000000000009e-152 or 8e11 < a

   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 z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]

      4. lower-*.f64 51.8

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(a \cdot t\right)}}{c} \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

   if -1.82000000000000009e-152 < a < 8e11

   1. Initial program 75.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

      3. lower-*.f64 39.6

         \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   5. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.82 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c}\\ \mathbf{elif}\;a \leq 800000000000:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{z \cdot c\_m} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* z c_m))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    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_m
    code = c_s * (b / (z * c_m))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (z * c_m))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(z * c_m)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (z * c_m));
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 b around inf

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

   3. lower-*.f64 28.9

      \[\leadsto \frac{b}{\color{blue}{z \cdot c}} \]

5. Applied rewrites 28.9%

   \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]
6. Add Preprocessing

Developer Target 1: 80.2% 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 2024223 
(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)))