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

Percentage Accurate: 79.3% → 92.5%

Time: 14.2s

Alternatives: 19

Speedup: 0.7×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{\mathsf{fma}\left(-9 \cdot x, \frac{y}{c\_m}, \frac{-b}{c\_m}\right)}{-z}\right)\\ \end{array} \end{array} \]

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.
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 (<= c_m 8.2e-50)
    (/ (/ (fma (* a (* z -4.0)) t (fma (* y x) 9.0 b)) z) c_m)
    (fma
     (* a -4.0)
     (/ t c_m)
     (/ (fma (* -9.0 x) (/ y c_m) (/ (- 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);
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 (c_m <= 8.2e-50) {
		tmp = (fma((a * (z * -4.0)), t, fma((y * x), 9.0, b)) / z) / c_m;
	} else {
		tmp = fma((a * -4.0), (t / c_m), (fma((-9.0 * x), (y / c_m), (-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])
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 (c_m <= 8.2e-50)
		tmp = Float64(Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(Float64(y * x), 9.0, b)) / z) / c_m);
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(fma(Float64(-9.0 * x), Float64(y / c_m), Float64(Float64(-b) / c_m)) / Float64(-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.
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[c$95$m, 8.2e-50], N[(N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(N[(N[(-9.0 * x), $MachinePrecision] * N[(y / c$95$m), $MachinePrecision] + N[((-b) / c$95$m), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 8.19999999999999971e-50

   1. Initial program 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   if 8.19999999999999971e-50 < c

   1. Initial program 64.3%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 89.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{\mathsf{fma}\left(-9 \cdot x, \frac{y}{c}, \frac{-b}{c}\right)}{-z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot -4\right)\\ t_2 := \frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\frac{z \cdot c\_m}{\mathsf{fma}\left(t\_1, t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(t\_1, t, b\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{b}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (* z -4.0)))
        (t_2 (/ (- b (- (* (* (* 4.0 z) t) a) (* (* 9.0 x) y))) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (/ 1.0 (/ (* z c_m) (fma t_1 t (fma (* y x) 9.0 b))))
      (if (<= t_2 4e+29)
        (/ (/ (fma (* 9.0 y) x (fma (* (* t a) -4.0) z b)) z) c_m)
        (if (<= t_2 INFINITY)
          (/ (fma (* 9.0 x) y (fma t_1 t b)) (* z c_m))
          (fma (* a -4.0) (/ t c_m) (/ b (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 * (z * -4.0);
	double t_2 = (b - ((((4.0 * z) * t) * a) - ((9.0 * x) * y))) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = 1.0 / ((z * c_m) / fma(t_1, t, fma((y * x), 9.0, b)));
	} else if (t_2 <= 4e+29) {
		tmp = (fma((9.0 * y), x, fma(((t * a) * -4.0), z, b)) / z) / c_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((9.0 * x), y, fma(t_1, t, b)) / (z * c_m);
	} else {
		tmp = fma((a * -4.0), (t / c_m), (b / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(a * Float64(z * -4.0))
	t_2 = Float64(Float64(b - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - Float64(Float64(9.0 * x) * y))) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(1.0 / Float64(Float64(z * c_m) / fma(t_1, t, fma(Float64(y * x), 9.0, b))));
	elseif (t_2 <= 4e+29)
		tmp = Float64(Float64(fma(Float64(9.0 * y), x, fma(Float64(Float64(t * a) * -4.0), z, b)) / z) / c_m);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(9.0 * x), y, fma(t_1, t, b)) / Float64(z * c_m));
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(b / Float64(z * c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(1.0 / N[(N[(z * c$95$m), $MachinePrecision] / N[(t$95$1 * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+29], N[(N[(N[(N[(9.0 * y), $MachinePrecision] * x + N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(t$95$1 * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(z * c$95$m), $MachinePrecision]), $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)) < -5e45

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.1

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

   4. Applied rewrites 90.1%

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

   if -5e45 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 3.99999999999999966e29

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 90.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if 3.99999999999999966e29 < (/.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 86.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 86.0%

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

   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 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 76.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\frac{z \cdot c}{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := a \cdot \left(z \cdot -4\right)\\ t_2 := \frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{z \cdot c\_m} \cdot \mathsf{fma}\left(t\_1, t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(t\_1, t, b\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{b}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (* z -4.0)))
        (t_2 (/ (- b (- (* (* (* 4.0 z) t) a) (* (* 9.0 x) y))) (* z c_m))))
   (*
    c_s
    (if (<= t_2 -5e+45)
      (* (/ 1.0 (* z c_m)) (fma t_1 t (fma (* y x) 9.0 b)))
      (if (<= t_2 4e+29)
        (/ (/ (fma (* 9.0 y) x (fma (* (* t a) -4.0) z b)) z) c_m)
        (if (<= t_2 INFINITY)
          (/ (fma (* 9.0 x) y (fma t_1 t b)) (* z c_m))
          (fma (* a -4.0) (/ t c_m) (/ b (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 * (z * -4.0);
	double t_2 = (b - ((((4.0 * z) * t) * a) - ((9.0 * x) * y))) / (z * c_m);
	double tmp;
	if (t_2 <= -5e+45) {
		tmp = (1.0 / (z * c_m)) * fma(t_1, t, fma((y * x), 9.0, b));
	} else if (t_2 <= 4e+29) {
		tmp = (fma((9.0 * y), x, fma(((t * a) * -4.0), z, b)) / z) / c_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((9.0 * x), y, fma(t_1, t, b)) / (z * c_m);
	} else {
		tmp = fma((a * -4.0), (t / c_m), (b / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(a * Float64(z * -4.0))
	t_2 = Float64(Float64(b - Float64(Float64(Float64(Float64(4.0 * z) * t) * a) - Float64(Float64(9.0 * x) * y))) / Float64(z * c_m))
	tmp = 0.0
	if (t_2 <= -5e+45)
		tmp = Float64(Float64(1.0 / Float64(z * c_m)) * fma(t_1, t, fma(Float64(y * x), 9.0, b)));
	elseif (t_2 <= 4e+29)
		tmp = Float64(Float64(fma(Float64(9.0 * y), x, fma(Float64(Float64(t * a) * -4.0), z, b)) / z) / c_m);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(9.0 * x), y, fma(t_1, t, b)) / Float64(z * c_m));
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(b / Float64(z * c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - N[(N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+45], N[(N[(1.0 / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+29], N[(N[(N[(N[(9.0 * y), $MachinePrecision] * x + N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(t$95$1 * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(z * c$95$m), $MachinePrecision]), $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)) < -5e45

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 91.1

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 90.0%

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

   if -5e45 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 3.99999999999999966e29

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 90.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if 3.99999999999999966e29 < (/.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 86.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 86.0%

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

   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 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 76.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{z \cdot c} \cdot \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c\_m}\\ t_2 := \frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{b}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (- (* (* (* 4.0 z) t) a) (* (* 9.0 x) y))) (* z c_m)))
        (t_2 (/ (fma (* 9.0 x) y (fma (* a (* z -4.0)) t b)) (* z c_m))))
   (*
    c_s
    (if (<= t_1 -5e+45)
      t_2
      (if (<= t_1 4e+29)
        (/ (/ (fma (* 9.0 y) x (fma (* (* t a) -4.0) z b)) z) c_m)
        (if (<= t_1 INFINITY)
          t_2
          (fma (* a -4.0) (/ t c_m) (/ b (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 - ((((4.0 * z) * t) * a) - ((9.0 * x) * y))) / (z * c_m);
	double t_2 = fma((9.0 * x), y, fma((a * (z * -4.0)), t, b)) / (z * c_m);
	double tmp;
	if (t_1 <= -5e+45) {
		tmp = t_2;
	} else if (t_1 <= 4e+29) {
		tmp = (fma((9.0 * y), x, fma(((t * a) * -4.0), z, b)) / z) / c_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma((a * -4.0), (t / c_m), (b / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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(Float64(Float64(4.0 * z) * t) * a) - Float64(Float64(9.0 * x) * y))) / Float64(z * c_m))
	t_2 = Float64(fma(Float64(9.0 * x), y, fma(Float64(a * Float64(z * -4.0)), t, b)) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= -5e+45)
		tmp = t_2;
	elseif (t_1 <= 4e+29)
		tmp = Float64(Float64(fma(Float64(9.0 * y), x, fma(Float64(Float64(t * a) * -4.0), z, b)) / z) / c_m);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(b / Float64(z * c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+45], t$95$2, If[LessEqual[t$95$1, 4e+29], N[(N[(N[(N[(9.0 * y), $MachinePrecision] * x + N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(z * c$95$m), $MachinePrecision]), $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)) < -5e45 or 3.99999999999999966e29 < (/.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 89.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 88.1%

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

   if -5e45 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 3.99999999999999966e29

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 90.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 76.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9 \cdot y, x, \mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)\right)}{z}}{c}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c\_m}\\ t_2 := \frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{b}{z \cdot c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (- (* (* (* 4.0 z) t) a) (* (* 9.0 x) y))) (* z c_m)))
        (t_2 (/ (fma (* 9.0 x) y (fma (* a (* z -4.0)) t b)) (* z c_m))))
   (*
    c_s
    (if (<= t_1 -2e-242)
      t_2
      (if (<= t_1 0.0)
        (/ (/ (fma (* y x) 9.0 b) c_m) z)
        (if (<= t_1 INFINITY)
          t_2
          (fma (* a -4.0) (/ t c_m) (/ b (* z c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 - ((((4.0 * z) * t) * a) - ((9.0 * x) * y))) / (z * c_m);
	double t_2 = fma((9.0 * x), y, fma((a * (z * -4.0)), t, b)) / (z * c_m);
	double tmp;
	if (t_1 <= -2e-242) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma((y * x), 9.0, b) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma((a * -4.0), (t / c_m), (b / (z * c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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(Float64(Float64(4.0 * z) * t) * a) - Float64(Float64(9.0 * x) * y))) / Float64(z * c_m))
	t_2 = Float64(fma(Float64(9.0 * x), y, fma(Float64(a * Float64(z * -4.0)), t, b)) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= -2e-242)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(b / Float64(z * c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision] - N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-242], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(z * c$95$m), $MachinePrecision]), $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)) < -2e-242 or -0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied rewrites 88.9%

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

   if -2e-242 < (/.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 55.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

   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 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 76.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq -2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b - \left(\left(\left(4 \cdot z\right) \cdot t\right) \cdot a - \left(9 \cdot x\right) \cdot y\right)}{z \cdot c} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, \mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}\\ t_2 := \left(9 \cdot x\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c\_m}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c\_m}, \frac{b}{z \cdot c\_m}\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c\_m} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \end{array} \]

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.
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 (* y x) 9.0 b) c_m) z)) (t_2 (* (* 9.0 x) y)))
   (*
    c_s
    (if (<= t_2 -5e+200)
      (* (/ x z) (/ (* 9.0 y) c_m))
      (if (<= t_2 -5e-51)
        t_1
        (if (<= t_2 4e-156)
          (fma (* a -4.0) (/ t c_m) (/ b (* z c_m)))
          (if (<= t_2 4e+210) t_1 (* (* (/ y c_m) 9.0) (/ x 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);
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((y * x), 9.0, b) / c_m) / z;
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -5e+200) {
		tmp = (x / z) * ((9.0 * y) / c_m);
	} else if (t_2 <= -5e-51) {
		tmp = t_1;
	} else if (t_2 <= 4e-156) {
		tmp = fma((a * -4.0), (t / c_m), (b / (z * c_m)));
	} else if (t_2 <= 4e+210) {
		tmp = t_1;
	} else {
		tmp = ((y / c_m) * 9.0) * (x / 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])
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(fma(Float64(y * x), 9.0, b) / c_m) / z)
	t_2 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_2 <= -5e+200)
		tmp = Float64(Float64(x / z) * Float64(Float64(9.0 * y) / c_m));
	elseif (t_2 <= -5e-51)
		tmp = t_1;
	elseif (t_2 <= 4e-156)
		tmp = fma(Float64(a * -4.0), Float64(t / c_m), Float64(b / Float64(z * c_m)));
	elseif (t_2 <= 4e+210)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / c_m) * 9.0) * Float64(x / 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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+200], N[(N[(x / z), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-51], t$95$1, If[LessEqual[t$95$2, 4e-156], N[(N[(a * -4.0), $MachinePrecision] * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+210], t$95$1, N[(N[(N[(y / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000019e200

   1. Initial program 67.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 84.0%

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

   if -5.00000000000000019e200 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000004e-51 or 4.00000000000000016e-156 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99999999999999971e210

   1. Initial program 85.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if -5.00000000000000004e-51 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.00000000000000016e-156

   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 b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 87.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 90.3%

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

   if 3.99999999999999971e210 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 63.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq -5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot -4, \frac{t}{c}, \frac{b}{z \cdot c}\right)\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 4 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ t_2 := \frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-82}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+210}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c\_m} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)) (t_2 (/ (/ (fma (* y x) 9.0 b) c_m) z)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ x z) (/ (* 9.0 y) c_m))
      (if (<= t_1 -4e-82)
        t_2
        (if (<= t_1 5e-149)
          (/ (fma (* a (* z -4.0)) t b) (* z c_m))
          (if (<= t_1 4e+210) t_2 (* (* (/ y c_m) 9.0) (/ x 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);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * x) * y;
	double t_2 = (fma((y * x), 9.0, b) / c_m) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / z) * ((9.0 * y) / c_m);
	} else if (t_1 <= -4e-82) {
		tmp = t_2;
	} else if (t_1 <= 5e-149) {
		tmp = fma((a * (z * -4.0)), t, b) / (z * c_m);
	} else if (t_1 <= 4e+210) {
		tmp = t_2;
	} else {
		tmp = ((y / c_m) * 9.0) * (x / 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])
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(9.0 * x) * y)
	t_2 = Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / z) * Float64(Float64(9.0 * y) / c_m));
	elseif (t_1 <= -4e-82)
		tmp = t_2;
	elseif (t_1 <= 5e-149)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, b) / Float64(z * c_m));
	elseif (t_1 <= 4e+210)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / c_m) * 9.0) * Float64(x / 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.
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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / z), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-82], t$95$2, If[LessEqual[t$95$1, 5e-149], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+210], t$95$2, N[(N[(N[(y / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 42.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4e-82 or 4.99999999999999968e-149 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99999999999999971e210

   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 a around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -4e-82 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999968e-149

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 76.3%

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

   if 3.99999999999999971e210 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 63.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq -4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 4 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ t_2 := \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+228}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c\_m} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)) (t_2 (/ (fma (* y x) 9.0 b) (* z c_m))))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ x z) (/ (* 9.0 y) c_m))
      (if (<= t_1 -5e-16)
        t_2
        (if (<= t_1 2e-199)
          (/ (fma (* (* t a) -4.0) z b) (* z c_m))
          (if (<= t_1 5e+228) t_2 (* (* (/ y c_m) 9.0) (/ x 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);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * x) * y;
	double t_2 = fma((y * x), 9.0, b) / (z * c_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x / z) * ((9.0 * y) / c_m);
	} else if (t_1 <= -5e-16) {
		tmp = t_2;
	} else if (t_1 <= 2e-199) {
		tmp = fma(((t * a) * -4.0), z, b) / (z * c_m);
	} else if (t_1 <= 5e+228) {
		tmp = t_2;
	} else {
		tmp = ((y / c_m) * 9.0) * (x / 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])
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(9.0 * x) * y)
	t_2 = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x / z) * Float64(Float64(9.0 * y) / c_m));
	elseif (t_1 <= -5e-16)
		tmp = t_2;
	elseif (t_1 <= 2e-199)
		tmp = Float64(fma(Float64(Float64(t * a) * -4.0), z, b) / Float64(z * c_m));
	elseif (t_1 <= 5e+228)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y / c_m) * 9.0) * Float64(x / 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.
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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x / z), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-16], t$95$2, If[LessEqual[t$95$1, 2e-199], N[(N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+228], t$95$2, N[(N[(N[(y / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 42.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000004e-16 or 1.99999999999999996e-199 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5e228

   1. Initial program 83.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -5.0000000000000004e-16 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999996e-199

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if 5e228 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{z} \cdot \frac{9 \cdot y}{c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t \cdot a\right) \cdot -4, z, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\left(\frac{y}{c\_m} \cdot x\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c\_m} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (* (* (/ y c_m) x) (/ 9.0 z))
      (if (<= t_1 2e-199)
        (/ (fma (* a (* z -4.0)) t b) (* z c_m))
        (if (<= t_1 5e+228)
          (/ (fma (* y x) 9.0 b) (* z c_m))
          (* (* (/ y c_m) 9.0) (/ x 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);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * x) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = ((y / c_m) * x) * (9.0 / z);
	} else if (t_1 <= 2e-199) {
		tmp = fma((a * (z * -4.0)), t, b) / (z * c_m);
	} else if (t_1 <= 5e+228) {
		tmp = fma((y * x), 9.0, b) / (z * c_m);
	} else {
		tmp = ((y / c_m) * 9.0) * (x / 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])
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(9.0 * x) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(Float64(Float64(y / c_m) * x) * Float64(9.0 / z));
	elseif (t_1 <= 2e-199)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, b) / Float64(z * c_m));
	elseif (t_1 <= 5e+228)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(y / c_m) * 9.0) * Float64(x / 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.
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[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(y / c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(9.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-199], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+228], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 21.4

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

   7. Applied rewrites 21.4%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999996e-199

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 74.4%

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

   if 1.99999999999999996e-199 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5e228

   1. Initial program 82.4%

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

   3. Taylor expanded in a 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 69.9

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

   5. Applied rewrites 69.9%

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

   if 5e228 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\left(\frac{y}{c} \cdot x\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{z \cdot c\_m} \cdot x\right) \cdot 9\\ t_2 := \left(9 \cdot x\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t \cdot a}{c\_m} \cdot -4\\ \mathbf{elif}\;t\_2 \leq 10:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (* z c_m)) x) 9.0)) (t_2 (* (* 9.0 x) y)))
   (*
    c_s
    (if (<= t_2 -1e-14)
      t_1
      (if (<= t_2 0.0)
        (* (/ (* t a) c_m) -4.0)
        (if (<= t_2 10.0) (/ b (* z c_m)) t_1))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 / (z * c_m)) * x) * 9.0;
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -1e-14) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = ((t * a) / c_m) * -4.0;
	} else if (t_2 <= 10.0) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((y / (z * c_m)) * x) * 9.0d0
    t_2 = (9.0d0 * x) * y
    if (t_2 <= (-1d-14)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = ((t * a) / c_m) * (-4.0d0)
    else if (t_2 <= 10.0d0) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
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 = ((y / (z * c_m)) * x) * 9.0;
	double t_2 = (9.0 * x) * y;
	double tmp;
	if (t_2 <= -1e-14) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = ((t * a) / c_m) * -4.0;
	} else if (t_2 <= 10.0) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[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 = ((y / (z * c_m)) * x) * 9.0
	t_2 = (9.0 * x) * y
	tmp = 0
	if t_2 <= -1e-14:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = ((t * a) / c_m) * -4.0
	elif t_2 <= 10.0:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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(y / Float64(z * c_m)) * x) * 9.0)
	t_2 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_2 <= -1e-14)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(t * a) / c_m) * -4.0);
	elseif (t_2 <= 10.0)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
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 = ((y / (z * c_m)) * x) * 9.0;
	t_2 = (9.0 * x) * y;
	tmp = 0.0;
	if (t_2 <= -1e-14)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = ((t * a) / c_m) * -4.0;
	elseif (t_2 <= 10.0)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 9.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -1e-14], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$2, 10.0], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.99999999999999999e-15 or 10 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if -9.99999999999999999e-15 < (*.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 a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   if 0.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 10

   1. Initial program 84.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 65.5

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

   5. Applied rewrites 65.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{y}{z \cdot c} \cdot x\right) \cdot 9\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 0:\\ \;\;\;\;\frac{t \cdot a}{c} \cdot -4\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 10:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{z \cdot c} \cdot x\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.46 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9 \cdot y}{z \cdot c\_m}, x, \mathsf{fma}\left(\frac{a}{c\_m} \cdot -4, t, \frac{b}{z \cdot c\_m}\right)\right)\\ \end{array} \end{array} \]

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.
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 (<= c_m 1.46e-98)
    (/ (/ (fma (* a (* z -4.0)) t (fma (* y x) 9.0 b)) z) c_m)
    (fma
     (/ (* 9.0 y) (* z c_m))
     x
     (fma (* (/ a c_m) -4.0) t (/ b (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 (c_m <= 1.46e-98) {
		tmp = (fma((a * (z * -4.0)), t, fma((y * x), 9.0, b)) / z) / c_m;
	} else {
		tmp = fma(((9.0 * y) / (z * c_m)), x, fma(((a / c_m) * -4.0), t, (b / (z * c_m))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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 (c_m <= 1.46e-98)
		tmp = Float64(Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(Float64(y * x), 9.0, b)) / z) / c_m);
	else
		tmp = fma(Float64(Float64(9.0 * y) / Float64(z * c_m)), x, fma(Float64(Float64(a / c_m) * -4.0), t, Float64(b / Float64(z * c_m))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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[c$95$m, 1.46e-98], N[(N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(9.0 * y), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.46000000000000007e-98

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   if 1.46000000000000007e-98 < c

   1. Initial program 66.7%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 87.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.46 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9 \cdot y}{z \cdot c}, x, \mathsf{fma}\left(\frac{a}{c} \cdot -4, t, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c\_m} \cdot 9, x, \mathsf{fma}\left(\frac{a}{c\_m} \cdot -4, t, \frac{b}{z \cdot c\_m}\right)\right)\\ \end{array} \end{array} \]

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.
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 (<= c_m 1.4e-98)
    (/ (/ (fma (* a (* z -4.0)) t (fma (* y x) 9.0 b)) z) c_m)
    (fma
     (* (/ y (* z c_m)) 9.0)
     x
     (fma (* (/ a c_m) -4.0) t (/ b (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 (c_m <= 1.4e-98) {
		tmp = (fma((a * (z * -4.0)), t, fma((y * x), 9.0, b)) / z) / c_m;
	} else {
		tmp = fma(((y / (z * c_m)) * 9.0), x, fma(((a / c_m) * -4.0), t, (b / (z * c_m))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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 (c_m <= 1.4e-98)
		tmp = Float64(Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(Float64(y * x), 9.0, b)) / z) / c_m);
	else
		tmp = fma(Float64(Float64(y / Float64(z * c_m)) * 9.0), x, fma(Float64(Float64(a / c_m) * -4.0), t, Float64(b / Float64(z * c_m))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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[c$95$m, 1.4e-98], N[(N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * x + N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.3999999999999999e-98

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   if 1.3999999999999999e-98 < c

   1. Initial program 66.7%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 87.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c} \cdot 9, x, \mathsf{fma}\left(\frac{a}{c} \cdot -4, t, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, t \cdot -4, \frac{\frac{b - \left(-9 \cdot x\right) \cdot y}{c\_m}}{z}\right)\\ \end{array} \end{array} \]

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.
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 (<= c_m 3.5e-22)
    (/ (/ 1.0 (/ z (fma (* a (* z -4.0)) t (fma (* 9.0 x) y b)))) c_m)
    (fma (/ a c_m) (* t -4.0) (/ (/ (- b (* (* -9.0 x) y)) 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);
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 (c_m <= 3.5e-22) {
		tmp = (1.0 / (z / fma((a * (z * -4.0)), t, fma((9.0 * x), y, b)))) / c_m;
	} else {
		tmp = fma((a / c_m), (t * -4.0), (((b - ((-9.0 * x) * y)) / 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])
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 (c_m <= 3.5e-22)
		tmp = Float64(Float64(1.0 / Float64(z / fma(Float64(a * Float64(z * -4.0)), t, fma(Float64(9.0 * x), y, b)))) / c_m);
	else
		tmp = fma(Float64(a / c_m), Float64(t * -4.0), Float64(Float64(Float64(b - Float64(Float64(-9.0 * x) * y)) / 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.
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[c$95$m, 3.5e-22], N[(N[(1.0 / N[(z / N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision] + N[(N[(N[(b - N[(N[(-9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 3.50000000000000005e-22

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 78.8

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if 3.50000000000000005e-22 < c

   1. Initial program 63.6%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 87.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(9 \cdot x, y, b\right)\right)}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c}, t \cdot -4, \frac{\frac{b - \left(-9 \cdot x\right) \cdot y}{c}}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 2.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, t \cdot -4, \frac{\frac{b - \left(-9 \cdot x\right) \cdot y}{c\_m}}{z}\right)\\ \end{array} \end{array} \]

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.
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 (<= c_m 2.3e-51)
    (/ (/ (fma (* a (* z -4.0)) t (fma (* y x) 9.0 b)) z) c_m)
    (fma (/ a c_m) (* t -4.0) (/ (/ (- b (* (* -9.0 x) y)) 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);
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 (c_m <= 2.3e-51) {
		tmp = (fma((a * (z * -4.0)), t, fma((y * x), 9.0, b)) / z) / c_m;
	} else {
		tmp = fma((a / c_m), (t * -4.0), (((b - ((-9.0 * x) * y)) / 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])
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 (c_m <= 2.3e-51)
		tmp = Float64(Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(Float64(y * x), 9.0, b)) / z) / c_m);
	else
		tmp = fma(Float64(a / c_m), Float64(t * -4.0), Float64(Float64(Float64(b - Float64(Float64(-9.0 * x) * y)) / 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.
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[c$95$m, 2.3e-51], N[(N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision] + N[(N[(N[(b - N[(N[(-9.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 2.30000000000000002e-51

   1. Initial program 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

   if 2.30000000000000002e-51 < c

   1. Initial program 64.3%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 87.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(y \cdot x, 9, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c}, t \cdot -4, \frac{\frac{b - \left(-9 \cdot x\right) \cdot y}{c}}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c\_m}{\left(t \cdot -4\right) \cdot a}}\\ \end{array} \end{array} \]

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.
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 (<= z -1.22e+54)
    (/ (* (* t a) -4.0) c_m)
    (if (<= z 4e+58)
      (/ (fma (* y x) 9.0 b) (* z c_m))
      (/ 1.0 (/ c_m (* (* t -4.0) a)))))))

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);
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 (z <= -1.22e+54) {
		tmp = ((t * a) * -4.0) / c_m;
	} else if (z <= 4e+58) {
		tmp = fma((y * x), 9.0, b) / (z * c_m);
	} else {
		tmp = 1.0 / (c_m / ((t * -4.0) * a));
	}
	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])
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 (z <= -1.22e+54)
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	elseif (z <= 4e+58)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c_m));
	else
		tmp = Float64(1.0 / Float64(c_m / Float64(Float64(t * -4.0) * a)));
	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.
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[z, -1.22e+54], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 4e+58], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m / N[(N[(t * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.22e54

   1. Initial program 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -1.22e54 < z < 3.99999999999999978e58

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 3.99999999999999978e58 < z

   1. Initial program 46.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 61.5

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

   7. Applied rewrites 61.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.5

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

   9. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{\left(t \cdot -4\right) \cdot a}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a}{c\_m} \cdot -4\\ \end{array} \end{array} \]

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.
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 (<= z -1.22e+54)
    (/ (* (* t a) -4.0) c_m)
    (if (<= z 4e+58)
      (/ (fma (* y x) 9.0 b) (* z c_m))
      (* (/ (* t a) c_m) -4.0)))))

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);
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 (z <= -1.22e+54) {
		tmp = ((t * a) * -4.0) / c_m;
	} else if (z <= 4e+58) {
		tmp = fma((y * x), 9.0, b) / (z * c_m);
	} else {
		tmp = ((t * a) / c_m) * -4.0;
	}
	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])
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 (z <= -1.22e+54)
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	elseif (z <= 4e+58)
		tmp = Float64(fma(Float64(y * x), 9.0, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) / c_m) * -4.0);
	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.
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[z, -1.22e+54], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 4e+58], N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.22e54

   1. Initial program 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -1.22e54 < z < 3.99999999999999978e58

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 3.99999999999999978e58 < z

   1. Initial program 46.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. *-commutative N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

      4. lower-*.f64 61.5

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{t \cdot a}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c\_m}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a}{c\_m} \cdot -4\\ \end{array} \end{array} \]

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.
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 (<= z -1.1e-56)
    (/ (* (* t a) -4.0) c_m)
    (if (<= z 1.9e+17) (/ b (* z c_m)) (* (/ (* t a) c_m) -4.0)))))

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);
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 (z <= -1.1e-56) {
		tmp = ((t * a) * -4.0) / c_m;
	} else if (z <= 1.9e+17) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) / c_m) * -4.0;
	}
	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.
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 (z <= (-1.1d-56)) then
        tmp = ((t * a) * (-4.0d0)) / c_m
    else if (z <= 1.9d+17) then
        tmp = b / (z * c_m)
    else
        tmp = ((t * a) / c_m) * (-4.0d0)
    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;
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 (z <= -1.1e-56) {
		tmp = ((t * a) * -4.0) / c_m;
	} else if (z <= 1.9e+17) {
		tmp = b / (z * c_m);
	} else {
		tmp = ((t * a) / c_m) * -4.0;
	}
	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])
[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 z <= -1.1e-56:
		tmp = ((t * a) * -4.0) / c_m
	elif z <= 1.9e+17:
		tmp = b / (z * c_m)
	else:
		tmp = ((t * a) / c_m) * -4.0
	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])
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 (z <= -1.1e-56)
		tmp = Float64(Float64(Float64(t * a) * -4.0) / c_m);
	elseif (z <= 1.9e+17)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = Float64(Float64(Float64(t * a) / c_m) * -4.0);
	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])){:}
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 (z <= -1.1e-56)
		tmp = ((t * a) * -4.0) / c_m;
	elseif (z <= 1.9e+17)
		tmp = b / (z * c_m);
	else
		tmp = ((t * a) / c_m) * -4.0;
	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.
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[z, -1.1e-56], N[(N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 1.9e+17], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.10000000000000002e-56

   1. Initial program 71.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}}{c}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right)}}{c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot t\right) \cdot -4}}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot t\right) \cdot -4}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot a\right)} \cdot -4}{c} \]

      4. lower-*.f64 50.9

         \[\leadsto \frac{\color{blue}{\left(t \cdot a\right)} \cdot -4}{c} \]

   7. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{\left(t \cdot a\right) \cdot -4}}{c} \]

   if -1.10000000000000002e-56 < z < 1.9e17

   1. Initial program 96.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 55.2

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   if 1.9e17 < z

   1. Initial program 54.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. *-commutative N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

      4. lower-*.f64 60.0

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{t \cdot a}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(t \cdot a\right) \cdot -4}{c}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{t \cdot a}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (/ (* t a) c_m) -4.0)))
   (* c_s (if (<= z -1.1e-56) t_1 (if (<= z 1.9e+17) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((t * a) / c_m) * -4.0;
	double tmp;
	if (z <= -1.1e-56) {
		tmp = t_1;
	} else if (z <= 1.9e+17) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t * a) / c_m) * (-4.0d0)
    if (z <= (-1.1d-56)) then
        tmp = t_1
    else if (z <= 1.9d+17) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((t * a) / c_m) * -4.0;
	double tmp;
	if (z <= -1.1e-56) {
		tmp = t_1;
	} else if (z <= 1.9e+17) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((t * a) / c_m) * -4.0
	tmp = 0
	if z <= -1.1e-56:
		tmp = t_1
	elif z <= 1.9e+17:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(t * a) / c_m) * -4.0)
	tmp = 0.0
	if (z <= -1.1e-56)
		tmp = t_1;
	elseif (z <= 1.9e+17)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((t * a) / c_m) * -4.0;
	tmp = 0.0;
	if (z <= -1.1e-56)
		tmp = t_1;
	elseif (z <= 1.9e+17)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.1e-56], t$95$1, If[LessEqual[z, 1.9e+17], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000002e-56 or 1.9e17 < z

   1. Initial program 62.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. *-commutative N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

      4. lower-*.f64 55.5

         \[\leadsto -4 \cdot \frac{\color{blue}{t \cdot a}}{c} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{t \cdot a}{c}} \]

   if -1.10000000000000002e-56 < z < 1.9e17

   1. Initial program 96.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 55.2

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{t \cdot a}{c} \cdot -4\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{z \cdot c\_m} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* z c_m))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    code = c_s * (b / (z * c_m))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (z * c_m))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(z * c_m)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (z * c_m));
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   2. lower-*.f64 37.9

      \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

5. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

6. Final simplification 37.9%

   \[\leadsto \frac{b}{z \cdot c} \]
7. Add Preprocessing

Developer Target 1: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024248 
(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)))