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

Percentage Accurate: 80.0% → 88.8%

Time: 13.4s

Alternatives: 14

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: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 0.98:\\ \;\;\;\;\frac{b + \left(\left(9 \cdot x\right) \cdot y - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c\_m} \cdot 9, x, \mathsf{fma}\left(-4 \cdot \frac{a}{c\_m}, t, \frac{b}{z \cdot c\_m}\right)\right)\\ \end{array} \end{array} \]

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 0.98)
    (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))
    (fma
     (* (/ y (* z c_m)) 9.0)
     x
     (fma (* -4.0 (/ a c_m)) 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);
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 <= 0.98) {
		tmp = (b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m);
	} else {
		tmp = fma(((y / (z * c_m)) * 9.0), x, fma((-4.0 * (a / c_m)), 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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 0.98)
		tmp = Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m));
	else
		tmp = fma(Float64(Float64(y / Float64(z * c_m)) * 9.0), x, fma(Float64(-4.0 * Float64(a / c_m)), 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 0.98], N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * x + N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 0.97999999999999998

   1. Initial program 82.8%

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

   if 0.97999999999999998 < c

   1. Initial program 69.4%

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

   3. Taylor expanded in 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 87.8%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 0.98:\\ \;\;\;\;\frac{b + \left(\left(9 \cdot x\right) \cdot y - a \cdot \left(t \cdot \left(4 \cdot z\right)\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c} \cdot 9, x, \mathsf{fma}\left(-4 \cdot \frac{a}{c}, t, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.4× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))))
   (*
    c_s
    (if (<= t_1 INFINITY)
      t_1
      (* (fma (/ (/ (/ (* y x) z) c_m) a) 9.0 (* (/ t c_m) -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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(((((y * x) / z) / c_m) / a), 9.0, ((t / c_m) * -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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(y * x) / z) / c_m) / a), 9.0, Float64(Float64(t / c_m) * -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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision] / a), $MachinePrecision] * 9.0 + N[(N[(t / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

   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

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.0%

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

4. Final simplification 85.8%

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

Alternative 3: 85.2% accurate, 0.4× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))))
   (*
    c_s
    (if (<= t_1 INFINITY)
      t_1
      (* (fma (* (/ x a) 9.0) (/ y (* z c_m)) (* (/ t c_m) -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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(((x / a) * 9.0), (y / (z * c_m)), ((t / c_m) * -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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(Float64(x / a) * 9.0), Float64(y / Float64(z * c_m)), Float64(Float64(t / c_m) * -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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(N[(x / a), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(t / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

   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

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 73.7%

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

4. Final simplification 85.8%

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

Alternative 4: 84.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

   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

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.3%

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

4. Final simplification 85.1%

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

Alternative 5: 84.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

   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

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

4. Final simplification 85.1%

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

Alternative 6: 84.9% accurate, 0.5× speedup

Math

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

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))))
   (*
    c_s
    (if (<= t_1 INFINITY) t_1 (fma (* -4.0 (/ a c_m)) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma((-4.0 * (a / c_m)), 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])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(Float64(-4.0 * Float64(a / c_m)), 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 69.2

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

   7. Applied rewrites 69.2%

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

4. Final simplification 85.4%

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

Alternative 7: 85.8% accurate, 0.5× speedup

Math

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

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<=
       (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))
       INFINITY)
    (/ (fma (* 9.0 x) y (fma (* (* a t) -4.0) z b)) (* z c_m))
    (fma (* -4.0 (/ a c_m)) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (((b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m)) <= ((double) INFINITY)) {
		tmp = fma((9.0 * x), y, fma(((a * t) * -4.0), z, b)) / (z * c_m);
	} else {
		tmp = fma((-4.0 * (a / c_m)), 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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m)) <= Inf)
		tmp = Float64(fma(Float64(9.0 * x), y, fma(Float64(Float64(a * t) * -4.0), z, b)) / Float64(z * c_m));
	else
		tmp = fma(Float64(-4.0 * Float64(a / c_m)), 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 86.1%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 69.2

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

   7. Applied rewrites 69.2%

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

4. Final simplification 84.6%

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

Alternative 8: 84.8% accurate, 0.5× speedup

Math

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

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.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<=
       (/ (+ b (- (* (* 9.0 x) y) (* a (* t (* 4.0 z))))) (* z c_m))
       INFINITY)
    (/ (fma (* 9.0 x) y (fma (* (* -4.0 z) a) t b)) (* z c_m))
    (fma (* -4.0 (/ a c_m)) 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (((b + (((9.0 * x) * y) - (a * (t * (4.0 * z))))) / (z * c_m)) <= ((double) INFINITY)) {
		tmp = fma((9.0 * x), y, fma(((-4.0 * z) * a), t, b)) / (z * c_m);
	} else {
		tmp = fma((-4.0 * (a / c_m)), 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])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (Float64(Float64(b + Float64(Float64(Float64(9.0 * x) * y) - Float64(a * Float64(t * Float64(4.0 * z))))) / Float64(z * c_m)) <= Inf)
		tmp = Float64(fma(Float64(9.0 * x), y, fma(Float64(Float64(-4.0 * z) * a), t, b)) / Float64(z * c_m));
	else
		tmp = fma(Float64(-4.0 * Float64(a / c_m)), 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.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[N[(N[(b + N[(N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision] - N[(a * N[(t * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(9.0 * x), $MachinePrecision] * y + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 69.2

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

   7. Applied rewrites 69.2%

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

4. Final simplification 84.6%

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

Alternative 9: 70.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot x\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{c\_m} \cdot 9\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{c\_m} \cdot x\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* 9.0 x) y)))
   (*
    c_s
    (if (<= t_1 -2e+144)
      (* (* (/ x c_m) 9.0) (/ y z))
      (if (<= t_1 5e-211)
        (/ (fma (* (* a t) -4.0) z b) (* z c_m))
        (if (<= t_1 5e+265)
          (/ (/ (fma (* y x) 9.0 b) c_m) z)
          (* (* (/ 9.0 c_m) x) (/ y z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * x) * y;
	double tmp;
	if (t_1 <= -2e+144) {
		tmp = ((x / c_m) * 9.0) * (y / z);
	} else if (t_1 <= 5e-211) {
		tmp = fma(((a * t) * -4.0), z, b) / (z * c_m);
	} else if (t_1 <= 5e+265) {
		tmp = (fma((y * x), 9.0, b) / c_m) / z;
	} else {
		tmp = ((9.0 / c_m) * x) * (y / z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(9.0 * x) * y)
	tmp = 0.0
	if (t_1 <= -2e+144)
		tmp = Float64(Float64(Float64(x / c_m) * 9.0) * Float64(y / z));
	elseif (t_1 <= 5e-211)
		tmp = Float64(fma(Float64(Float64(a * t) * -4.0), z, b) / Float64(z * c_m));
	elseif (t_1 <= 5e+265)
		tmp = Float64(Float64(fma(Float64(y * x), 9.0, b) / c_m) / z);
	else
		tmp = Float64(Float64(Float64(9.0 / c_m) * x) * Float64(y / z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(9.0 * x), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+144], N[(N[(N[(x / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-211], N[(N[(N[(N[(a * t), $MachinePrecision] * -4.0), $MachinePrecision] * z + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+265], N[(N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(9.0 / c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000005e144

   1. Initial program 71.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 22.7%

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

   9. Taylor expanded in y around inf

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 75.9

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

   11. Applied rewrites 75.9%

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

   if -2.00000000000000005e144 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e-211

   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 73.7

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

   5. Applied rewrites 73.7%

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

   if 5.0000000000000002e-211 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e265

   1. Initial program 85.3%

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

   3. Taylor expanded in 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 71.4

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

   5. Applied rewrites 71.4%

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

   if 5.0000000000000002e265 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 16.2

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   7. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

   10. Applied rewrites 83.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot x\right) \cdot y \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{c} \cdot 9\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot t\right) \cdot -4, z, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(9 \cdot x\right) \cdot y \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{c} \cdot x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-4 \cdot \frac{a}{c\_m}, t, \frac{b}{z \cdot c\_m}\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma (* -4.0 (/ a c_m)) t (/ b (* z c_m)))))
   (*
    c_s
    (if (<= z -1.7e+43)
      t_1
      (if (<= z 4.5e+81) (/ (fma (* 9.0 x) y b) (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((-4.0 * (a / c_m)), t, (b / (z * c_m)));
	double tmp;
	if (z <= -1.7e+43) {
		tmp = t_1;
	} else if (z <= 4.5e+81) {
		tmp = fma((9.0 * x), y, b) / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(Float64(-4.0 * Float64(a / c_m)), t, Float64(b / Float64(z * c_m)))
	tmp = 0.0
	if (z <= -1.7e+43)
		tmp = t_1;
	elseif (z <= 4.5e+81)
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(-4.0 * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.7e+43], t$95$1, If[LessEqual[z, 4.5e+81], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000006e43 or 4.50000000000000017e81 < z

   1. Initial program 56.3%

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

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -1.70000000000000006e43 < z < 4.50000000000000017e81

   1. Initial program 93.4%

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

   3. 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 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 81.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot \frac{a}{c}, t, \frac{b}{z \cdot c}\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4 \cdot \frac{a}{c}, t, \frac{b}{z \cdot c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+190}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (/ (* a t) c_m) -4.0)))
   (*
    c_s
    (if (<= z -2.1e+47)
      t_1
      (if (<= z 1.8e+190) (/ (fma (* 9.0 x) y b) (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (z <= -2.1e+47) {
		tmp = t_1;
	} else if (z <= 1.8e+190) {
		tmp = fma((9.0 * x), y, b) / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(a * t) / c_m) * -4.0)
	tmp = 0.0
	if (z <= -2.1e+47)
		tmp = t_1;
	elseif (z <= 1.8e+190)
		tmp = Float64(fma(Float64(9.0 * x), y, b) / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -2.1e+47], t$95$1, If[LessEqual[z, 1.8e+190], N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e47 or 1.79999999999999989e190 < z

   1. Initial program 51.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -2.1e47 < z < 1.79999999999999989e190

   1. Initial program 89.3%

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

   3. Taylor expanded in 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 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 76.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+190}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{x}{z \cdot c\_m} \cdot y\right) \cdot 9\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (/ (* a t) c_m) -4.0)))
   (*
    c_s
    (if (<= z -1.06e+45)
      t_1
      (if (<= z 1.7e+43) (* (* (/ x (* z c_m)) y) 9.0) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (z <= -1.06e+45) {
		tmp = t_1;
	} else if (z <= 1.7e+43) {
		tmp = ((x / (z * c_m)) * y) * 9.0;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a * t) / c_m) * (-4.0d0)
    if (z <= (-1.06d+45)) then
        tmp = t_1
    else if (z <= 1.7d+43) then
        tmp = ((x / (z * c_m)) * y) * 9.0d0
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (z <= -1.06e+45) {
		tmp = t_1;
	} else if (z <= 1.7e+43) {
		tmp = ((x / (z * c_m)) * y) * 9.0;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((a * t) / c_m) * -4.0
	tmp = 0
	if z <= -1.06e+45:
		tmp = t_1
	elif z <= 1.7e+43:
		tmp = ((x / (z * c_m)) * y) * 9.0
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(a * t) / c_m) * -4.0)
	tmp = 0.0
	if (z <= -1.06e+45)
		tmp = t_1;
	elseif (z <= 1.7e+43)
		tmp = Float64(Float64(Float64(x / Float64(z * c_m)) * y) * 9.0);
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((a * t) / c_m) * -4.0;
	tmp = 0.0;
	if (z <= -1.06e+45)
		tmp = t_1;
	elseif (z <= 1.7e+43)
		tmp = ((x / (z * c_m)) * y) * 9.0;
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.06e+45], t$95$1, If[LessEqual[z, 1.7e+43], N[(N[(N[(x / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * 9.0), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.06e45 or 1.70000000000000006e43 < z

   1. Initial program 59.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 56.0

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

   5. Applied rewrites 56.0%

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

   if -1.06e45 < z < 1.70000000000000006e43

   1. Initial program 93.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 44.5

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   7. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.9

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

   8. Applied rewrites 50.9%

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

   10. Applied rewrites 55.9%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+45}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{x}{z \cdot c} \cdot y\right) \cdot 9\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (/ (* a t) c_m) -4.0)))
   (* c_s (if (<= z -5.6e+37) t_1 (if (<= z 4e+35) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (z <= -5.6e+37) {
		tmp = t_1;
	} else if (z <= 4e+35) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a * t) / c_m) * (-4.0d0)
    if (z <= (-5.6d+37)) then
        tmp = t_1
    else if (z <= 4d+35) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (z <= -5.6e+37) {
		tmp = t_1;
	} else if (z <= 4e+35) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((a * t) / c_m) * -4.0
	tmp = 0
	if z <= -5.6e+37:
		tmp = t_1
	elif z <= 4e+35:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(a * t) / c_m) * -4.0)
	tmp = 0.0
	if (z <= -5.6e+37)
		tmp = t_1;
	elseif (z <= 4e+35)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((a * t) / c_m) * -4.0;
	tmp = 0.0;
	if (z <= -5.6e+37)
		tmp = t_1;
	elseif (z <= 4e+35)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -5.6e+37], t$95$1, If[LessEqual[z, 4e+35], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5999999999999996e37 or 3.9999999999999999e35 < z

   1. Initial program 59.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 55.0

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

   5. Applied rewrites 55.0%

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

   if -5.5999999999999996e37 < z < 3.9999999999999999e35

   1. Initial program 94.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{z \cdot c\_m} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* z c_m))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    code = c_s * (b / (z * c_m))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (z * c_m));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (z * c_m))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(z * c_m)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (z * c_m));
end

Wolfram

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

Derivation

1. Initial program 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 34.9

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

5. Applied rewrites 34.9%

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

6. Final simplification 34.9%

   \[\leadsto \frac{b}{z \cdot c} \]
7. Add Preprocessing

Developer Target 1: 81.2% 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 2024244 
(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)))