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

Percentage Accurate: 80.0% → 88.8%

Time: 11.2s

Alternatives: 17

Speedup: 0.8×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, \frac{z}{c\_m}, \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{a}\right) \cdot a}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{c\_m}, \frac{y}{z}, \frac{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c\_m}}{x}\right) \cdot x\\ \end{array} \end{array} \end{array} \]

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 (/ (- b (- (* a (* t (* 4.0 z))) (* y (* 9.0 x)))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -5e-210)
      (/ (fma (* (* a t) -4.0) z (fma (* 9.0 x) y b)) (* c_m z))
      (if (<= t_1 2e-267)
        (/
         (* (fma (* -4.0 t) (/ z c_m) (/ (/ (fma (* y x) 9.0 b) c_m) a)) a)
         z)
        (if (<= t_1 4e+302)
          t_1
          (*
           (fma (/ 9.0 c_m) (/ y z) (/ (/ (fma (* -4.0 t) a (/ b z)) c_m) x))
           x)))))))

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 = (b - ((a * (t * (4.0 * z))) - (y * (9.0 * x)))) / (c_m * z);
	double tmp;
	if (t_1 <= -5e-210) {
		tmp = fma(((a * t) * -4.0), z, fma((9.0 * x), y, b)) / (c_m * z);
	} else if (t_1 <= 2e-267) {
		tmp = (fma((-4.0 * t), (z / c_m), ((fma((y * x), 9.0, b) / c_m) / a)) * a) / z;
	} else if (t_1 <= 4e+302) {
		tmp = t_1;
	} else {
		tmp = fma((9.0 / c_m), (y / z), ((fma((-4.0 * t), a, (b / z)) / c_m) / x)) * x;
	}
	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(b - Float64(Float64(a * Float64(t * Float64(4.0 * z))) - Float64(y * Float64(9.0 * x)))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -5e-210)
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, fma(Float64(9.0 * x), y, b)) / Float64(c_m * z));
	elseif (t_1 <= 2e-267)
		tmp = Float64(Float64(fma(Float64(-4.0 * t), Float64(z / c_m), Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / a)) * a) / z);
	elseif (t_1 <= 4e+302)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(9.0 / c_m), Float64(y / z), Float64(Float64(fma(Float64(-4.0 * t), a, Float64(b / z)) / c_m) / x)) * x);
	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[(b - N[(N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e-210], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-267], N[(N[(N[(N[(-4.0 * t), $MachinePrecision] * N[(z / c$95$m), $MachinePrecision] + N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 4e+302], t$95$1, N[(N[(N[(9.0 / c$95$m), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -5.0000000000000002e-210

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 91.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 91.8%

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

   if -5.0000000000000002e-210 < (/.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)) < 2e-267

   1. Initial program 42.9%

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

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

      4. associate-/r* N/A

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

      5. 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 93.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.3%

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

   if 2e-267 < (/.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)) < 4.0000000000000003e302

   1. Initial program 98.7%

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

   if 4.0000000000000003e302 < (/.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 55.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq 2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, \frac{z}{c}, \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{a}\right) \cdot a}{z}\\ \mathbf{elif}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{c}, \frac{y}{z}, \frac{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c}}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c\_m}{t}}\\ \end{array} \end{array} \end{array} \]

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 (/ (- b (- (* a (* t (* 4.0 z))) (* y (* 9.0 x)))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -4e-292)
      (/ (fma (* (* a t) -4.0) z (fma (* 9.0 x) y b)) (* c_m z))
      (if (<= t_1 0.0)
        (/ (/ (fma -4.0 (* (* t z) a) b) z) c_m)
        (if (<= t_1 INFINITY) t_1 (/ (* -4.0 a) (/ c_m t))))))))

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 = (b - ((a * (t * (4.0 * z))) - (y * (9.0 * x)))) / (c_m * z);
	double tmp;
	if (t_1 <= -4e-292) {
		tmp = fma(((a * t) * -4.0), z, fma((9.0 * x), y, b)) / (c_m * z);
	} else if (t_1 <= 0.0) {
		tmp = (fma(-4.0, ((t * z) * a), b) / z) / c_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (-4.0 * a) / (c_m / t);
	}
	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(b - Float64(Float64(a * Float64(t * Float64(4.0 * z))) - Float64(y * Float64(9.0 * x)))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -4e-292)
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, fma(Float64(9.0 * x), y, b)) / Float64(c_m * z));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / z) / c_m);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c_m / t));
	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[(b - N[(N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -4e-292], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(-4.0 * a), $MachinePrecision] / N[(c$95$m / t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -4.0000000000000002e-292

   1. Initial program 94.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 92.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if -4.0000000000000002e-292 < (/.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 22.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

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

   1. Initial program 86.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 66.3%

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

   13. Applied rewrites 66.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq -4 \cdot 10^{-292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c\_m}{t}}\\ \end{array} \end{array} \end{array} \]

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 (/ (- b (- (* a (* t (* 4.0 z))) (* y (* 9.0 x)))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e-202)
      (/ (fma (* (* a t) -4.0) z (fma (* 9.0 x) y b)) (* c_m z))
      (if (<= t_1 INFINITY)
        (/ (/ (fma (* (* -4.0 z) a) t (fma (* y x) 9.0 b)) c_m) z)
        (/ (* -4.0 a) (/ c_m t)))))))

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 = (b - ((a * (t * (4.0 * z))) - (y * (9.0 * x)))) / (c_m * z);
	double tmp;
	if (t_1 <= -2e-202) {
		tmp = fma(((a * t) * -4.0), z, fma((9.0 * x), y, b)) / (c_m * z);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (fma(((-4.0 * z) * a), t, fma((y * x), 9.0, b)) / c_m) / z;
	} else {
		tmp = (-4.0 * a) / (c_m / t);
	}
	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(b - Float64(Float64(a * Float64(t * Float64(4.0 * z))) - Float64(y * Float64(9.0 * x)))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e-202)
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, fma(Float64(9.0 * x), y, b)) / Float64(c_m * z));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(y * x), 9.0, b)) / c_m) / z);
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c_m / t));
	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[(b - N[(N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-202], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] / N[(c$95$m / t), $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-202

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 91.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 91.8%

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

   if -2.0000000000000001e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. 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 86.0%

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

   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. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 66.3%

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

   13. Applied rewrites 66.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b - \left(a \cdot \left(t \cdot \left(4 \cdot z\right)\right) - y \cdot \left(9 \cdot x\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{x}{c\_m \cdot z} \cdot \left(y \cdot 9\right)\\ t_2 := \frac{\frac{b}{c\_m}}{z}\\ t_3 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot t}{c\_m} \cdot -4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-235}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;t\_3 \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c\_m}{t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (/ x (* c_m z)) (* y 9.0)))
        (t_2 (/ (/ b c_m) z))
        (t_3 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_3 -1e+63)
      t_1
      (if (<= t_3 -5e-93)
        (* (/ (* a t) c_m) -4.0)
        (if (<= t_3 -1e-235)
          t_2
          (if (<= t_3 0.0)
            (* (* (/ a c_m) -4.0) t)
            (if (<= t_3 2.8e-129)
              t_2
              (if (<= t_3 2e+112) (/ (* -4.0 a) (/ c_m t)) 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 = (x / (c_m * z)) * (y * 9.0);
	double t_2 = (b / c_m) / z;
	double t_3 = y * (9.0 * x);
	double tmp;
	if (t_3 <= -1e+63) {
		tmp = t_1;
	} else if (t_3 <= -5e-93) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (t_3 <= -1e-235) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (t_3 <= 2.8e-129) {
		tmp = t_2;
	} else if (t_3 <= 2e+112) {
		tmp = (-4.0 * a) / (c_m / t);
	} 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) :: t_3
    real(8) :: tmp
    t_1 = (x / (c_m * z)) * (y * 9.0d0)
    t_2 = (b / c_m) / z
    t_3 = y * (9.0d0 * x)
    if (t_3 <= (-1d+63)) then
        tmp = t_1
    else if (t_3 <= (-5d-93)) then
        tmp = ((a * t) / c_m) * (-4.0d0)
    else if (t_3 <= (-1d-235)) then
        tmp = t_2
    else if (t_3 <= 0.0d0) then
        tmp = ((a / c_m) * (-4.0d0)) * t
    else if (t_3 <= 2.8d-129) then
        tmp = t_2
    else if (t_3 <= 2d+112) then
        tmp = ((-4.0d0) * a) / (c_m / t)
    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 = (x / (c_m * z)) * (y * 9.0);
	double t_2 = (b / c_m) / z;
	double t_3 = y * (9.0 * x);
	double tmp;
	if (t_3 <= -1e+63) {
		tmp = t_1;
	} else if (t_3 <= -5e-93) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (t_3 <= -1e-235) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (t_3 <= 2.8e-129) {
		tmp = t_2;
	} else if (t_3 <= 2e+112) {
		tmp = (-4.0 * a) / (c_m / t);
	} 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 = (x / (c_m * z)) * (y * 9.0)
	t_2 = (b / c_m) / z
	t_3 = y * (9.0 * x)
	tmp = 0
	if t_3 <= -1e+63:
		tmp = t_1
	elif t_3 <= -5e-93:
		tmp = ((a * t) / c_m) * -4.0
	elif t_3 <= -1e-235:
		tmp = t_2
	elif t_3 <= 0.0:
		tmp = ((a / c_m) * -4.0) * t
	elif t_3 <= 2.8e-129:
		tmp = t_2
	elif t_3 <= 2e+112:
		tmp = (-4.0 * a) / (c_m / t)
	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(x / Float64(c_m * z)) * Float64(y * 9.0))
	t_2 = Float64(Float64(b / c_m) / z)
	t_3 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_3 <= -1e+63)
		tmp = t_1;
	elseif (t_3 <= -5e-93)
		tmp = Float64(Float64(Float64(a * t) / c_m) * -4.0);
	elseif (t_3 <= -1e-235)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (t_3 <= 2.8e-129)
		tmp = t_2;
	elseif (t_3 <= 2e+112)
		tmp = Float64(Float64(-4.0 * a) / Float64(c_m / t));
	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 = (x / (c_m * z)) * (y * 9.0);
	t_2 = (b / c_m) / z;
	t_3 = y * (9.0 * x);
	tmp = 0.0;
	if (t_3 <= -1e+63)
		tmp = t_1;
	elseif (t_3 <= -5e-93)
		tmp = ((a * t) / c_m) * -4.0;
	elseif (t_3 <= -1e-235)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = ((a / c_m) * -4.0) * t;
	elseif (t_3 <= 2.8e-129)
		tmp = t_2;
	elseif (t_3 <= 2e+112)
		tmp = (-4.0 * a) / (c_m / t);
	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[(x / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$3, -1e+63], t$95$1, If[LessEqual[t$95$3, -5e-93], N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$3, -1e-235], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$3, 2.8e-129], t$95$2, If[LessEqual[t$95$3, 2e+112], N[(N[(-4.0 * a), $MachinePrecision] / N[(c$95$m / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000006e63 or 1.9999999999999999e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 70.1%

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

   if -1.00000000000000006e63 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999994e-93

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -4.99999999999999994e-93 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.9999999999999996e-236 or -0.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.7999999999999999e-129

   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. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 68.3%

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

   if -9.9999999999999996e-236 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -0.0

   1. Initial program 81.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 66.2%

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

   if 2.7999999999999999e-129 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.9999999999999999e112

   1. Initial program 81.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.3%

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

   13. Applied rewrites 56.3%

       \[\leadsto \frac{-4 \cdot a}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{c \cdot z} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 0:\\ \;\;\;\;\left(\frac{a}{c} \cdot -4\right) \cdot t\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{-4 \cdot a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{c \cdot z} \cdot \left(y \cdot 9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{x}{c\_m \cdot z} \cdot \left(y \cdot 9\right)\\ t_2 := \frac{\frac{b}{c\_m}}{z}\\ t_3 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot t}{c\_m} \cdot -4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-235}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;t\_3 \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (/ x (* c_m z)) (* y 9.0)))
        (t_2 (/ (/ b c_m) z))
        (t_3 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_3 -1e+63)
      t_1
      (if (<= t_3 -5e-93)
        (* (/ (* a t) c_m) -4.0)
        (if (<= t_3 -1e-235)
          t_2
          (if (<= t_3 0.0)
            (* (* (/ a c_m) -4.0) t)
            (if (<= t_3 2.8e-129)
              t_2
              (if (<= t_3 2e+112) (* (* -4.0 a) (/ t 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 = (x / (c_m * z)) * (y * 9.0);
	double t_2 = (b / c_m) / z;
	double t_3 = y * (9.0 * x);
	double tmp;
	if (t_3 <= -1e+63) {
		tmp = t_1;
	} else if (t_3 <= -5e-93) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (t_3 <= -1e-235) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (t_3 <= 2.8e-129) {
		tmp = t_2;
	} else if (t_3 <= 2e+112) {
		tmp = (-4.0 * a) * (t / 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) :: t_3
    real(8) :: tmp
    t_1 = (x / (c_m * z)) * (y * 9.0d0)
    t_2 = (b / c_m) / z
    t_3 = y * (9.0d0 * x)
    if (t_3 <= (-1d+63)) then
        tmp = t_1
    else if (t_3 <= (-5d-93)) then
        tmp = ((a * t) / c_m) * (-4.0d0)
    else if (t_3 <= (-1d-235)) then
        tmp = t_2
    else if (t_3 <= 0.0d0) then
        tmp = ((a / c_m) * (-4.0d0)) * t
    else if (t_3 <= 2.8d-129) then
        tmp = t_2
    else if (t_3 <= 2d+112) then
        tmp = ((-4.0d0) * a) * (t / 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 = (x / (c_m * z)) * (y * 9.0);
	double t_2 = (b / c_m) / z;
	double t_3 = y * (9.0 * x);
	double tmp;
	if (t_3 <= -1e+63) {
		tmp = t_1;
	} else if (t_3 <= -5e-93) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (t_3 <= -1e-235) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (t_3 <= 2.8e-129) {
		tmp = t_2;
	} else if (t_3 <= 2e+112) {
		tmp = (-4.0 * a) * (t / 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 = (x / (c_m * z)) * (y * 9.0)
	t_2 = (b / c_m) / z
	t_3 = y * (9.0 * x)
	tmp = 0
	if t_3 <= -1e+63:
		tmp = t_1
	elif t_3 <= -5e-93:
		tmp = ((a * t) / c_m) * -4.0
	elif t_3 <= -1e-235:
		tmp = t_2
	elif t_3 <= 0.0:
		tmp = ((a / c_m) * -4.0) * t
	elif t_3 <= 2.8e-129:
		tmp = t_2
	elif t_3 <= 2e+112:
		tmp = (-4.0 * a) * (t / 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(x / Float64(c_m * z)) * Float64(y * 9.0))
	t_2 = Float64(Float64(b / c_m) / z)
	t_3 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_3 <= -1e+63)
		tmp = t_1;
	elseif (t_3 <= -5e-93)
		tmp = Float64(Float64(Float64(a * t) / c_m) * -4.0);
	elseif (t_3 <= -1e-235)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (t_3 <= 2.8e-129)
		tmp = t_2;
	elseif (t_3 <= 2e+112)
		tmp = Float64(Float64(-4.0 * a) * Float64(t / 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 = (x / (c_m * z)) * (y * 9.0);
	t_2 = (b / c_m) / z;
	t_3 = y * (9.0 * x);
	tmp = 0.0;
	if (t_3 <= -1e+63)
		tmp = t_1;
	elseif (t_3 <= -5e-93)
		tmp = ((a * t) / c_m) * -4.0;
	elseif (t_3 <= -1e-235)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = ((a / c_m) * -4.0) * t;
	elseif (t_3 <= 2.8e-129)
		tmp = t_2;
	elseif (t_3 <= 2e+112)
		tmp = (-4.0 * a) * (t / 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[(x / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] * N[(y * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$3, -1e+63], t$95$1, If[LessEqual[t$95$3, -5e-93], N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$3, -1e-235], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$3, 2.8e-129], t$95$2, If[LessEqual[t$95$3, 2e+112], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000006e63 or 1.9999999999999999e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 70.1%

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

   if -1.00000000000000006e63 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999994e-93

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -4.99999999999999994e-93 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.9999999999999996e-236 or -0.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.7999999999999999e-129

   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. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 68.3%

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

   if -9.9999999999999996e-236 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -0.0

   1. Initial program 81.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 66.2%

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

   if 2.7999999999999999e-129 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.9999999999999999e112

   1. Initial program 81.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.3%

       \[\leadsto \left(-4 \cdot a\right) \cdot \frac{\color{blue}{t}}{c} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{c \cdot z} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 0:\\ \;\;\;\;\left(\frac{a}{c} \cdot -4\right) \cdot t\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{c \cdot z} \cdot \left(y \cdot 9\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.5× speedup

Math

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

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 (<=
       (/ (- b (- (* a (* t (* 4.0 z))) (* y (* 9.0 x)))) (* c_m z))
       INFINITY)
    (/ (fma (* (* a t) -4.0) z (fma (* 9.0 x) y b)) (* c_m z))
    (/ (* -4.0 a) (/ c_m t)))))

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 (((b - ((a * (t * (4.0 * z))) - (y * (9.0 * x)))) / (c_m * z)) <= ((double) INFINITY)) {
		tmp = fma(((a * t) * -4.0), z, fma((9.0 * x), y, b)) / (c_m * z);
	} else {
		tmp = (-4.0 * a) / (c_m / t);
	}
	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 (Float64(Float64(b - Float64(Float64(a * Float64(t * Float64(4.0 * z))) - Float64(y * Float64(9.0 * x)))) / Float64(c_m * z)) <= Inf)
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, fma(Float64(9.0 * x), y, b)) / Float64(c_m * z));
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c_m / t));
	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[N[(N[(b - N[(N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] / N[(c$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 88.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 66.3%

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

   13. Applied rewrites 66.2%

       \[\leadsto \frac{-4 \cdot a}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

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

Alternative 7: 85.3% accurate, 0.5× speedup

Math

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

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 (<=
       (/ (- b (- (* a (* t (* 4.0 z))) (* y (* 9.0 x)))) (* c_m z))
       INFINITY)
    (/ (fma (* y 9.0) x (fma (* -4.0 z) (* a t) b)) (* c_m z))
    (/ (* -4.0 a) (/ c_m t)))))

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 (((b - ((a * (t * (4.0 * z))) - (y * (9.0 * x)))) / (c_m * z)) <= ((double) INFINITY)) {
		tmp = fma((y * 9.0), x, fma((-4.0 * z), (a * t), b)) / (c_m * z);
	} else {
		tmp = (-4.0 * a) / (c_m / t);
	}
	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 (Float64(Float64(b - Float64(Float64(a * Float64(t * Float64(4.0 * z))) - Float64(y * Float64(9.0 * x)))) / Float64(c_m * z)) <= Inf)
		tmp = Float64(fma(Float64(y * 9.0), x, fma(Float64(-4.0 * z), Float64(a * t), b)) / Float64(c_m * z));
	else
		tmp = Float64(Float64(-4.0 * a) / Float64(c_m / t));
	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[N[(N[(b - N[(N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(-4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * a), $MachinePrecision] / N[(c$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 88.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 66.3%

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

   13. Applied rewrites 66.2%

       \[\leadsto \frac{-4 \cdot a}{\color{blue}{\frac{c}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

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

Alternative 8: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+225}:\\ \;\;\;\;\frac{y}{c\_m} \cdot \left(\frac{x}{z} \cdot 9\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c\_m} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e+225)
      (* (/ y c_m) (* (/ x z) 9.0))
      (if (<= t_1 -2e+48)
        (/ (/ (fma (* y x) 9.0 b) z) c_m)
        (if (<= t_1 1e+107)
          (/ (fma -4.0 (* (* t z) a) b) (* c_m z))
          (* (* (/ x c_m) 9.0) (/ y z))))))))

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 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -2e+225) {
		tmp = (y / c_m) * ((x / z) * 9.0);
	} else if (t_1 <= -2e+48) {
		tmp = (fma((y * x), 9.0, b) / z) / c_m;
	} else if (t_1 <= 1e+107) {
		tmp = fma(-4.0, ((t * z) * a), b) / (c_m * z);
	} else {
		tmp = ((x / c_m) * 9.0) * (y / z);
	}
	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(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e+225)
		tmp = Float64(Float64(y / c_m) * Float64(Float64(x / z) * 9.0));
	elseif (t_1 <= -2e+48)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / z) / c_m);
	elseif (t_1 <= 1e+107)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(Float64(x / c_m) * 9.0) * Float64(y / z));
	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[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+225], N[(N[(y / c$95$m), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+48], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+107], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.99999999999999986e225

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 82.4%

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

   if -1.99999999999999986e225 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000009e48

   1. Initial program 79.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -2.00000000000000009e48 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e106

   1. Initial program 82.5%

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

   3. 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 82.1

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

   11. Applied rewrites 82.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+225}:\\ \;\;\;\;\frac{y}{c} \cdot \left(\frac{x}{z} \cdot 9\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{z}}{c\_m} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c\_m} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -5e+270)
      (* (/ (/ x z) c_m) (* y 9.0))
      (if (<= t_1 -2e+48)
        (/ (fma (* y x) 9.0 b) (* c_m z))
        (if (<= t_1 1e+107)
          (/ (fma -4.0 (* (* t z) a) b) (* c_m z))
          (* (* (/ x c_m) 9.0) (/ y z))))))))

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 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -5e+270) {
		tmp = ((x / z) / c_m) * (y * 9.0);
	} else if (t_1 <= -2e+48) {
		tmp = fma((y * x), 9.0, b) / (c_m * z);
	} else if (t_1 <= 1e+107) {
		tmp = fma(-4.0, ((t * z) * a), b) / (c_m * z);
	} else {
		tmp = ((x / c_m) * 9.0) * (y / z);
	}
	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(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -5e+270)
		tmp = Float64(Float64(Float64(x / z) / c_m) * Float64(y * 9.0));
	elseif (t_1 <= -2e+48)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(c_m * z));
	elseif (t_1 <= 1e+107)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(Float64(x / c_m) * 9.0) * Float64(y / z));
	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[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+270], N[(N[(N[(x / z), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(y * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+48], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+107], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999976e270

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -4.99999999999999976e270 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000009e48

   1. Initial program 81.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -2.00000000000000009e48 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e106

   1. Initial program 82.5%

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

   3. 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 82.1

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

   11. Applied rewrites 82.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{z}}{c} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{z}}{c\_m} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(a \cdot z\right) \cdot t, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c\_m} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -5e+270)
      (* (/ (/ x z) c_m) (* y 9.0))
      (if (<= t_1 -2e+48)
        (/ (fma (* y x) 9.0 b) (* c_m z))
        (if (<= t_1 1e+107)
          (/ (fma -4.0 (* (* a z) t) b) (* c_m z))
          (* (* (/ x c_m) 9.0) (/ y z))))))))

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 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -5e+270) {
		tmp = ((x / z) / c_m) * (y * 9.0);
	} else if (t_1 <= -2e+48) {
		tmp = fma((y * x), 9.0, b) / (c_m * z);
	} else if (t_1 <= 1e+107) {
		tmp = fma(-4.0, ((a * z) * t), b) / (c_m * z);
	} else {
		tmp = ((x / c_m) * 9.0) * (y / z);
	}
	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(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -5e+270)
		tmp = Float64(Float64(Float64(x / z) / c_m) * Float64(y * 9.0));
	elseif (t_1 <= -2e+48)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(c_m * z));
	elseif (t_1 <= 1e+107)
		tmp = Float64(fma(-4.0, Float64(Float64(a * z) * t), b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(Float64(x / c_m) * 9.0) * Float64(y / z));
	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[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+270], N[(N[(N[(x / z), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(y * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+48], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+107], N[(N[(-4.0 * N[(N[(a * z), $MachinePrecision] * t), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999976e270

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -4.99999999999999976e270 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000009e48

   1. Initial program 81.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -2.00000000000000009e48 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999997e106

   1. Initial program 82.5%

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

   3. 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-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 75.0%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 74.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 82.1

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

   11. Applied rewrites 82.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{z}}{c} \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(a \cdot z\right) \cdot t, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{c} \cdot 9\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{t}{c\_m} \cdot \mathsf{fma}\left(-4, a, \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{t}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c\_m \cdot z}\\ \end{array} \end{array} \]

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 -9.8e-32)
    (* (/ t c_m) (fma -4.0 a (/ (/ (fma (* y x) 9.0 b) t) z)))
    (/ (fma (* (* a t) -4.0) z (fma (* 9.0 x) y b)) (* c_m z)))))

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 <= -9.8e-32) {
		tmp = (t / c_m) * fma(-4.0, a, ((fma((y * x), 9.0, b) / t) / z));
	} else {
		tmp = fma(((a * t) * -4.0), z, fma((9.0 * x), y, b)) / (c_m * z);
	}
	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 <= -9.8e-32)
		tmp = Float64(Float64(t / c_m) * fma(-4.0, a, Float64(Float64(fma(Float64(y * x), 9.0, b) / t) / z)));
	else
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, fma(Float64(9.0 * x), y, b)) / Float64(c_m * z));
	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, -9.8e-32], N[(N[(t / c$95$m), $MachinePrecision] * N[(-4.0 * a + N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -9.7999999999999996e-32

   1. Initial program 76.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 91.4%

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

   if -9.7999999999999996e-32 < t

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

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

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 81.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 81.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{t}{c} \cdot \mathsf{fma}\left(-4, a, \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{t}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right) \cdot \frac{t}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (fma -4.0 a (/ b (* t z))) (/ t c_m))))
   (*
    c_s
    (if (<= t -1.75e-19)
      t_1
      (if (<= t 8.2e-190) (/ (fma (* y x) 9.0 b) (* c_m z)) 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 = fma(-4.0, a, (b / (t * z))) * (t / c_m);
	double tmp;
	if (t <= -1.75e-19) {
		tmp = t_1;
	} else if (t <= 8.2e-190) {
		tmp = fma((y * x), 9.0, b) / (c_m * z);
	} 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(fma(-4.0, a, Float64(b / Float64(t * z))) * Float64(t / c_m))
	tmp = 0.0
	if (t <= -1.75e-19)
		tmp = t_1;
	elseif (t <= 8.2e-190)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(c_m * z));
	else
		tmp = t_1;
	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[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -1.75e-19], t$95$1, If[LessEqual[t, 8.2e-190], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.75000000000000008e-19 or 8.2000000000000004e-190 < t

   1. Initial program 75.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 86.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.1%

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

   if -1.75000000000000008e-19 < t < 8.2000000000000004e-190

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right) \cdot \frac{t}{c}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right) \cdot \frac{t}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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) c_m) -4.0)))
   (*
    c_s
    (if (<= z -1.2e+63)
      t_1
      (if (<= z 1.55e+138) (/ (fma (* y x) 9.0 b) (* c_m z)) 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) / c_m) * -4.0;
	double tmp;
	if (z <= -1.2e+63) {
		tmp = t_1;
	} else if (z <= 1.55e+138) {
		tmp = fma((y * x), 9.0, b) / (c_m * z);
	} 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(a * t) / c_m) * -4.0)
	tmp = 0.0
	if (z <= -1.2e+63)
		tmp = t_1;
	elseif (z <= 1.55e+138)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(c_m * z));
	else
		tmp = t_1;
	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[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.2e+63], t$95$1, If[LessEqual[z, 1.55e+138], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e63 or 1.5499999999999999e138 < z

   1. Initial program 50.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -1.2e63 < z < 1.5499999999999999e138

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c\_m}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \end{array} \end{array} \]

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 -5e+32)
    (* (* -4.0 a) (/ t c_m))
    (if (<= t 8.5e-80) (/ b (* c_m z)) (* (* (/ a c_m) -4.0) t)))))

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 <= -5e+32) {
		tmp = (-4.0 * a) * (t / c_m);
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} else {
		tmp = ((a / c_m) * -4.0) * t;
	}
	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 <= (-5d+32)) then
        tmp = ((-4.0d0) * a) * (t / c_m)
    else if (t <= 8.5d-80) then
        tmp = b / (c_m * z)
    else
        tmp = ((a / c_m) * (-4.0d0)) * t
    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 <= -5e+32) {
		tmp = (-4.0 * a) * (t / c_m);
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} else {
		tmp = ((a / c_m) * -4.0) * t;
	}
	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 <= -5e+32:
		tmp = (-4.0 * a) * (t / c_m)
	elif t <= 8.5e-80:
		tmp = b / (c_m * z)
	else:
		tmp = ((a / c_m) * -4.0) * t
	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 <= -5e+32)
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c_m));
	elseif (t <= 8.5e-80)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	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 <= -5e+32)
		tmp = (-4.0 * a) * (t / c_m);
	elseif (t <= 8.5e-80)
		tmp = b / (c_m * z);
	else
		tmp = ((a / c_m) * -4.0) * t;
	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, -5e+32], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-80], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.9999999999999997e32

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 23.3

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 63.0%

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

   if -4.9999999999999997e32 < t < 8.49999999999999939e-80

   1. Initial program 85.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   if 8.49999999999999939e-80 < t

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.7%

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

Alternative 15: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c\_m}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4}{c\_m} \cdot a\right) \cdot t\\ \end{array} \end{array} \]

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 -5e+32)
    (* (* -4.0 a) (/ t c_m))
    (if (<= t 8.5e-80) (/ b (* c_m z)) (* (* (/ -4.0 c_m) a) t)))))

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 <= -5e+32) {
		tmp = (-4.0 * a) * (t / c_m);
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} else {
		tmp = ((-4.0 / c_m) * a) * t;
	}
	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 <= (-5d+32)) then
        tmp = ((-4.0d0) * a) * (t / c_m)
    else if (t <= 8.5d-80) then
        tmp = b / (c_m * z)
    else
        tmp = (((-4.0d0) / c_m) * a) * t
    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 <= -5e+32) {
		tmp = (-4.0 * a) * (t / c_m);
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} else {
		tmp = ((-4.0 / c_m) * a) * t;
	}
	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 <= -5e+32:
		tmp = (-4.0 * a) * (t / c_m)
	elif t <= 8.5e-80:
		tmp = b / (c_m * z)
	else:
		tmp = ((-4.0 / c_m) * a) * t
	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 <= -5e+32)
		tmp = Float64(Float64(-4.0 * a) * Float64(t / c_m));
	elseif (t <= 8.5e-80)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = Float64(Float64(Float64(-4.0 / c_m) * a) * t);
	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 <= -5e+32)
		tmp = (-4.0 * a) * (t / c_m);
	elseif (t <= 8.5e-80)
		tmp = b / (c_m * z);
	else
		tmp = ((-4.0 / c_m) * a) * t;
	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, -5e+32], N[(N[(-4.0 * a), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-80], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 / c$95$m), $MachinePrecision] * a), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.9999999999999997e32

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 23.3

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 63.0%

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

   if -4.9999999999999997e32 < t < 8.49999999999999939e-80

   1. Initial program 85.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   if 8.49999999999999939e-80 < t

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 59.7%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\left(-4 \cdot a\right) \cdot \frac{t}{c}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4}{c} \cdot a\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(-4 \cdot a\right) \cdot \frac{t}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* -4.0 a) (/ t c_m))))
   (* c_s (if (<= t -5e+32) t_1 (if (<= t 8.5e-80) (/ b (* c_m z)) 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 = (-4.0 * a) * (t / c_m);
	double tmp;
	if (t <= -5e+32) {
		tmp = t_1;
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} 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 = ((-4.0d0) * a) * (t / c_m)
    if (t <= (-5d+32)) then
        tmp = t_1
    else if (t <= 8.5d-80) then
        tmp = b / (c_m * z)
    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 = (-4.0 * a) * (t / c_m);
	double tmp;
	if (t <= -5e+32) {
		tmp = t_1;
	} else if (t <= 8.5e-80) {
		tmp = b / (c_m * z);
	} 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 = (-4.0 * a) * (t / c_m)
	tmp = 0
	if t <= -5e+32:
		tmp = t_1
	elif t <= 8.5e-80:
		tmp = b / (c_m * z)
	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(-4.0 * a) * Float64(t / c_m))
	tmp = 0.0
	if (t <= -5e+32)
		tmp = t_1;
	elseif (t <= 8.5e-80)
		tmp = Float64(b / Float64(c_m * z));
	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 = (-4.0 * a) * (t / c_m);
	tmp = 0.0;
	if (t <= -5e+32)
		tmp = t_1;
	elseif (t <= 8.5e-80)
		tmp = b / (c_m * z);
	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[(-4.0 * a), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -5e+32], t$95$1, If[LessEqual[t, 8.5e-80], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999997e32 or 8.49999999999999939e-80 < t

   1. Initial program 74.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. div-add N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 60.4%

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

   if -4.9999999999999997e32 < t < 8.49999999999999939e-80

   1. Initial program 85.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{c\_m \cdot z} \end{array} \]

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 (* c_m z))))

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 / (c_m * z));
}

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 / (c_m * z))
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 / (c_m * z));
}

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 / (c_m * z))

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(c_m * z)))
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 / (c_m * z));
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[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 36.5

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

5. Applied rewrites 36.5%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024298 
(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)))