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

Percentage Accurate: 79.2% → 88.1%

Time: 14.8s

Alternatives: 19

Speedup: 0.9×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% 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: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{c\_m} \cdot -4, t, \frac{b}{z \cdot c\_m}\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-1}{c\_m}}{\frac{-z}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)\right)}}\\ \mathbf{elif}\;c\_m \leq 5.3 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c\_m} \cdot 9, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{9}{c\_m}, \frac{t\_1}{y}\right) \cdot y\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma (* (/ a c_m) -4.0) t (/ b (* z c_m)))))
   (*
    c_s
    (if (<= c_m 4e-19)
      (/ (/ -1.0 c_m) (/ (- z) (fma (* y 9.0) x (fma (* -4.0 (* a t)) z b))))
      (if (<= c_m 5.3e+181)
        (fma (* (/ y (* z c_m)) 9.0) x t_1)
        (* (fma (/ x z) (/ 9.0 c_m) (/ t_1 y)) y))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(((a / c_m) * -4.0), t, (b / (z * c_m)));
	double tmp;
	if (c_m <= 4e-19) {
		tmp = (-1.0 / c_m) / (-z / fma((y * 9.0), x, fma((-4.0 * (a * t)), z, b)));
	} else if (c_m <= 5.3e+181) {
		tmp = fma(((y / (z * c_m)) * 9.0), x, t_1);
	} else {
		tmp = fma((x / z), (9.0 / c_m), (t_1 / y)) * y;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = fma(Float64(Float64(a / c_m) * -4.0), t, Float64(b / Float64(z * c_m)))
	tmp = 0.0
	if (c_m <= 4e-19)
		tmp = Float64(Float64(-1.0 / c_m) / Float64(Float64(-z) / fma(Float64(y * 9.0), x, fma(Float64(-4.0 * Float64(a * t)), z, b))));
	elseif (c_m <= 5.3e+181)
		tmp = fma(Float64(Float64(y / Float64(z * c_m)) * 9.0), x, t_1);
	else
		tmp = Float64(fma(Float64(x / z), Float64(9.0 / c_m), Float64(t_1 / y)) * y);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[c$95$m, 4e-19], N[(N[(-1.0 / c$95$m), $MachinePrecision] / N[((-z) / N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c$95$m, 5.3e+181], N[(N[(N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(9.0 / c$95$m), $MachinePrecision] + N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if c < 3.9999999999999999e-19

   1. Initial program 80.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 87.3%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 87.5

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 91.0%

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

   if 3.9999999999999999e-19 < c < 5.2999999999999996e181

   1. Initial program 70.9%

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

   3. Taylor expanded in b around 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 94.9%

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

   if 5.2999999999999996e181 < c

   1. Initial program 49.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-1}{c}}{\frac{-z}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)\right)}}\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c} \cdot 9, x, \mathsf{fma}\left(\frac{a}{c} \cdot -4, t, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{9}{c}, \frac{\mathsf{fma}\left(\frac{a}{c} \cdot -4, t, \frac{b}{z \cdot c}\right)}{y}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot t\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m} \leq 10^{+290}:\\ \;\;\;\;\frac{\frac{-1}{c\_m}}{\frac{-z}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(t\_1, z, b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, t\_1\right)}{c\_m}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* -4.0 (* a t))))
   (*
    c_s
    (if (<=
         (/ (+ (- (* (* x 9.0) y) (* (* (* 4.0 z) t) a)) b) (* z c_m))
         1e+290)
      (/ (/ -1.0 c_m) (/ (- z) (fma (* y 9.0) x (fma t_1 z b))))
      (/ (fma (* (/ x z) y) 9.0 t_1) c_m)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -4.0 * (a * t);
	double tmp;
	if ((((((x * 9.0) * y) - (((4.0 * z) * t) * a)) + b) / (z * c_m)) <= 1e+290) {
		tmp = (-1.0 / c_m) / (-z / fma((y * 9.0), x, fma(t_1, z, b)));
	} else {
		tmp = fma(((x / z) * y), 9.0, t_1) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(-4.0 * Float64(a * t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(4.0 * z) * t) * a)) + b) / Float64(z * c_m)) <= 1e+290)
		tmp = Float64(Float64(-1.0 / c_m) / Float64(Float64(-z) / fma(Float64(y * 9.0), x, fma(t_1, z, b))));
	else
		tmp = Float64(fma(Float64(Float64(x / z) * y), 9.0, t_1) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], 1e+290], N[(N[(-1.0 / c$95$m), $MachinePrecision] / N[((-z) / N[(N[(y * 9.0), $MachinePrecision] * x + N[(t$95$1 * z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 9.0 + t$95$1), $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)) < 1.00000000000000006e290

   1. Initial program 84.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 86.3%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 87.4

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 93.7%

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

   if 1.00000000000000006e290 < (/.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 51.8%

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

   3. Taylor expanded in b around 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 48.7

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

   5. Applied rewrites 48.7%

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

   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 52.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 76.3%

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

   13. Applied rewrites 81.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 10^{+290}:\\ \;\;\;\;\frac{\frac{-1}{c}}{\frac{-z}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\frac{x}{z}, -9, \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{-y}\right) \cdot y\right) \cdot \frac{-1}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -90000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1
         (*
          (* (fma (/ x z) -9.0 (/ (fma (* -4.0 a) t (/ b z)) (- y))) y)
          (/ -1.0 c_m))))
   (*
    c_s
    (if (<= z -90000000.0)
      t_1
      (if (<= z 3e-5)
        (/ (+ (- (* (* x 9.0) y) (* (* (* 4.0 z) t) a)) b) (* z c_m))
        t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (fma((x / z), -9.0, (fma((-4.0 * a), t, (b / z)) / -y)) * y) * (-1.0 / c_m);
	double tmp;
	if (z <= -90000000.0) {
		tmp = t_1;
	} else if (z <= 3e-5) {
		tmp = ((((x * 9.0) * y) - (((4.0 * z) * t) * a)) + b) / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(fma(Float64(x / z), -9.0, Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / Float64(-y))) * y) * Float64(-1.0 / c_m))
	tmp = 0.0
	if (z <= -90000000.0)
		tmp = t_1;
	elseif (z <= 3e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(4.0 * z) * t) * a)) + 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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(N[(x / z), $MachinePrecision] * -9.0 + N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * N[(-1.0 / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -90000000.0], t$95$1, If[LessEqual[z, 3e-5], N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9e7 or 3.00000000000000008e-5 < z

   1. Initial program 57.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. times-frac N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -9e7 < z < 3.00000000000000008e-5

   1. Initial program 96.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -90000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x}{z}, -9, \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{-y}\right) \cdot y\right) \cdot \frac{-1}{c}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x}{z}, -9, \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{-y}\right) \cdot y\right) \cdot \frac{-1}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c\_m}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (/ (fma (* x y) 9.0 b) z) c_m)
      (if (<= t_1 1e+54)
        (/ (fma (* -4.0 a) t (/ b z)) c_m)
        (/ (fma (* (/ x z) y) 9.0 (* -4.0 (* a t))) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = (fma((x * y), 9.0, b) / z) / c_m;
	} else if (t_1 <= 1e+54) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = fma(((x / z) * y), 9.0, (-4.0 * (a * t))) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(Float64(fma(Float64(x * y), 9.0, b) / z) / c_m);
	elseif (t_1 <= 1e+54)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = Float64(fma(Float64(Float64(x / z) * y), 9.0, Float64(-4.0 * Float64(a * t))) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+54], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 9.0 + N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.0000000000000001e54

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 76.4%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if 1.0000000000000001e54 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   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 28.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 78.4%

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

   13. Applied rewrites 83.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{x \cdot y}{z} \cdot 9\right)}{c\_m}\\ \end{array} \end{array} \end{array} \]

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.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (/ (fma (* x y) 9.0 b) z) c_m)
      (if (<= t_1 1e+54)
        (/ (fma (* -4.0 a) t (/ b z)) c_m)
        (/ (fma (* a t) -4.0 (* (/ (* x y) z) 9.0)) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = (fma((x * y), 9.0, b) / z) / c_m;
	} else if (t_1 <= 1e+54) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = fma((a * t), -4.0, (((x * y) / z) * 9.0)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(Float64(fma(Float64(x * y), 9.0, b) / z) / c_m);
	elseif (t_1 <= 1e+54)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(Float64(Float64(x * y) / z) * 9.0)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e+54], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.0000000000000001e54

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 76.4%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if 1.0000000000000001e54 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 67.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{x \cdot y}{z} \cdot 9\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.7× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 4e-19)
    (/ (/ -1.0 c_m) (/ (- z) (fma (* y 9.0) x (fma (* -4.0 (* a t)) z b))))
    (fma
     (* (/ y (* z c_m)) 9.0)
     x
     (fma (* (/ a c_m) -4.0) t (/ b (* z c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 4e-19) {
		tmp = (-1.0 / c_m) / (-z / fma((y * 9.0), x, fma((-4.0 * (a * t)), z, b)));
	} else {
		tmp = fma(((y / (z * c_m)) * 9.0), x, fma(((a / c_m) * -4.0), t, (b / (z * c_m))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 4e-19)
		tmp = Float64(Float64(-1.0 / c_m) / Float64(Float64(-z) / fma(Float64(y * 9.0), x, fma(Float64(-4.0 * Float64(a * t)), z, b))));
	else
		tmp = fma(Float64(Float64(y / Float64(z * c_m)) * 9.0), x, fma(Float64(Float64(a / c_m) * -4.0), t, Float64(b / Float64(z * c_m))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 4e-19], N[(N[(-1.0 / c$95$m), $MachinePrecision] / N[((-z) / N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] * x + N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t + N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 3.9999999999999999e-19

   1. Initial program 80.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 87.3%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 87.5

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 91.0%

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

   if 3.9999999999999999e-19 < c

   1. Initial program 62.1%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-*l/ N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 87.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{-1}{c}}{\frac{-z}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot c} \cdot 9, x, \mathsf{fma}\left(\frac{a}{c} \cdot -4, t, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c\_m}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}}{c\_m}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -2.15e+170)
    (/ (fma (* (/ x z) y) 9.0 (* -4.0 (* a t))) c_m)
    (if (<= z -1e-37)
      (/ (/ (fma (* (* -4.0 z) a) t (fma (* x y) 9.0 b)) z) c_m)
      (if (<= z 2.2e+154)
        (/ (+ (- (* (* x 9.0) y) (* (* (* 4.0 z) t) a)) b) (* z c_m))
        (/ (fma (* -4.0 a) t (/ b z)) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -2.15e+170) {
		tmp = fma(((x / z) * y), 9.0, (-4.0 * (a * t))) / c_m;
	} else if (z <= -1e-37) {
		tmp = (fma(((-4.0 * z) * a), t, fma((x * y), 9.0, b)) / z) / c_m;
	} else if (z <= 2.2e+154) {
		tmp = ((((x * 9.0) * y) - (((4.0 * z) * t) * a)) + b) / (z * c_m);
	} else {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -2.15e+170)
		tmp = Float64(fma(Float64(Float64(x / z) * y), 9.0, Float64(-4.0 * Float64(a * t))) / c_m);
	elseif (z <= -1e-37)
		tmp = Float64(Float64(fma(Float64(Float64(-4.0 * z) * a), t, fma(Float64(x * y), 9.0, b)) / z) / c_m);
	elseif (z <= 2.2e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(4.0 * z) * t) * a)) + b) / Float64(z * c_m));
	else
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -2.15e+170], N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 9.0 + N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, -1e-37], N[(N[(N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 2.2e+154], N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -2.1499999999999999e170

   1. Initial program 22.5%

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

   3. Taylor expanded in b around 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 38.2

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

   5. Applied rewrites 38.2%

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

   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 56.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 72.2%

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

   13. Applied rewrites 81.1%

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

   if -2.1499999999999999e170 < z < -1.00000000000000007e-37

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 90.8%

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

   if -1.00000000000000007e-37 < z < 2.2000000000000001e154

   1. Initial program 94.8%

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

   if 2.2000000000000001e154 < z

   1. Initial program 42.3%

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

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 61.0%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(x \cdot y, 9, b\right)\right)}{z}}{c}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c\_m} \cdot x\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (/ (fma (* x y) 9.0 b) z) c_m)
      (if (<= t_1 5e+145)
        (/ (fma (* -4.0 a) t (/ b z)) c_m)
        (* (/ y z) (* (/ 9.0 c_m) x)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = (fma((x * y), 9.0, b) / z) / c_m;
	} else if (t_1 <= 5e+145) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = (y / z) * ((9.0 / c_m) * x);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(Float64(fma(Float64(x * y), 9.0, b) / z) / c_m);
	elseif (t_1 <= 5e+145)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(9.0 / c_m) * x));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e+145], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(9.0 / c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999967e145

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 75.4%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if 4.99999999999999967e145 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 70.6%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 70.6

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   8. 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 78.6

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 78.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z}}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c\_m} \cdot x\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (/ (fma (* x y) 9.0 b) c_m) z)
      (if (<= t_1 5e+145)
        (/ (fma (* -4.0 a) t (/ b z)) c_m)
        (* (/ y z) (* (/ 9.0 c_m) x)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = (fma((x * y), 9.0, b) / c_m) / z;
	} else if (t_1 <= 5e+145) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = (y / z) * ((9.0 / c_m) * x);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(Float64(fma(Float64(x * y), 9.0, b) / c_m) / z);
	elseif (t_1 <= 5e+145)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(9.0 / c_m) * x));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+145], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(9.0 / c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 69.8

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

   5. Applied rewrites 69.8%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999967e145

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 75.4%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if 4.99999999999999967e145 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 70.6%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 70.6

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   8. 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 78.6

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 78.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{c}}{z}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c\_m} \cdot x\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (fma (* x 9.0) y b) (* z c_m))
      (if (<= t_1 5e+145)
        (/ (fma (* -4.0 a) t (/ b z)) c_m)
        (* (/ y z) (* (/ 9.0 c_m) x)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = fma((x * 9.0), y, b) / (z * c_m);
	} else if (t_1 <= 5e+145) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = (y / z) * ((9.0 / c_m) * x);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(fma(Float64(x * 9.0), y, b) / Float64(z * c_m));
	elseif (t_1 <= 5e+145)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(9.0 / c_m) * x));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+145], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(9.0 / c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 66.1%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999967e145

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 75.4%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if 4.99999999999999967e145 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 70.6%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 70.6

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   8. 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 78.6

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 78.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c\_m} \cdot x\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 -1e+31)
      (/ (fma (* x 9.0) y b) (* z c_m))
      (if (<= t_1 2e+135)
        (/ (fma (* -4.0 (* a t)) z b) (* z c_m))
        (* (/ y z) (* (/ 9.0 c_m) x)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -1e+31) {
		tmp = fma((x * 9.0), y, b) / (z * c_m);
	} else if (t_1 <= 2e+135) {
		tmp = fma((-4.0 * (a * t)), z, b) / (z * c_m);
	} else {
		tmp = (y / z) * ((9.0 / c_m) * x);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= -1e+31)
		tmp = Float64(fma(Float64(x * 9.0), y, b) / Float64(z * c_m));
	elseif (t_1 <= 2e+135)
		tmp = Float64(fma(Float64(-4.0 * Float64(a * t)), z, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(9.0 / c_m) * x));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -1e+31], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+135], N[(N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] * z + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(9.0 / c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 66.1%

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

   if -9.9999999999999996e30 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999992e135

   1. Initial program 79.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   if 1.99999999999999992e135 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 62.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. 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 68.7%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 68.7

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

      11. lift-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-+l+ N/A

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   8. 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 75.8

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

   9. Applied rewrites 75.8%

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

   11. Applied rewrites 75.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot \left(a \cdot t\right), z, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{9}{c} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c\_m}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -9.4e-13)
    (/ (fma (* (/ x z) y) 9.0 (* -4.0 (* a t))) c_m)
    (if (<= z 2.2e+154)
      (/ (+ (- (* (* x 9.0) y) (* (* (* 4.0 z) t) a)) b) (* z c_m))
      (/ (fma (* -4.0 a) t (/ b z)) c_m)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -9.4e-13) {
		tmp = fma(((x / z) * y), 9.0, (-4.0 * (a * t))) / c_m;
	} else if (z <= 2.2e+154) {
		tmp = ((((x * 9.0) * y) - (((4.0 * z) * t) * a)) + b) / (z * c_m);
	} else {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -9.4e-13)
		tmp = Float64(fma(Float64(Float64(x / z) * y), 9.0, Float64(-4.0 * Float64(a * t))) / c_m);
	elseif (z <= 2.2e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(4.0 * z) * t) * a)) + b) / Float64(z * c_m));
	else
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -9.4e-13], N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 9.0 + N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 2.2e+154], N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(4.0 * z), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -9.4000000000000003e-13

   1. Initial program 49.5%

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

   3. Taylor expanded in b around 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 58.9

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

   5. Applied rewrites 58.9%

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

   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 56.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 75.9%

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

   13. Applied rewrites 80.0%

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

   if -9.4000000000000003e-13 < z < 2.2000000000000001e154

   1. Initial program 94.9%

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

   if 2.2000000000000001e154 < z

   1. Initial program 42.3%

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

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 61.0%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(4 \cdot z\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.059:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c\_m}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -0.059)
    (/ (fma (* (/ x z) y) 9.0 (* -4.0 (* a t))) c_m)
    (if (<= z 4e+154)
      (/ (fma (* x 9.0) y (fma (* (* -4.0 z) a) t b)) (* z c_m))
      (/ (fma (* -4.0 a) t (/ b z)) c_m)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -0.059) {
		tmp = fma(((x / z) * y), 9.0, (-4.0 * (a * t))) / c_m;
	} else if (z <= 4e+154) {
		tmp = fma((x * 9.0), y, fma(((-4.0 * z) * a), t, b)) / (z * c_m);
	} else {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -0.059)
		tmp = Float64(fma(Float64(Float64(x / z) * y), 9.0, Float64(-4.0 * Float64(a * t))) / c_m);
	elseif (z <= 4e+154)
		tmp = Float64(fma(Float64(x * 9.0), y, fma(Float64(Float64(-4.0 * z) * a), t, b)) / Float64(z * c_m));
	else
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -0.059], N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 9.0 + N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[z, 4e+154], N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(N[(N[(-4.0 * z), $MachinePrecision] * a), $MachinePrecision] * t + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -0.058999999999999997

   1. Initial program 48.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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 56.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 75.5%

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

   13. Applied rewrites 79.7%

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

   if -0.058999999999999997 < z < 4.00000000000000015e154

   1. Initial program 95.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 92.6%

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

   if 4.00000000000000015e154 < z

   1. Initial program 42.3%

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

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

      4. associate-+l- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 61.0%

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

   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. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

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

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.059:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 9, -4 \cdot \left(a \cdot t\right)\right)}{c}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, b\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{a \cdot t}{c\_m} \cdot -4\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (<= a -0.0045)
    (* (/ (* a t) c_m) -4.0)
    (if (<= a 1.7e+68)
      (/ (fma (* x y) 9.0 b) (* z c_m))
      (/ (* -4.0 (* a t)) c_m)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 (a <= -0.0045) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (a <= 1.7e+68) {
		tmp = fma((x * y), 9.0, b) / (z * c_m);
	} else {
		tmp = (-4.0 * (a * t)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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 (a <= -0.0045)
		tmp = Float64(Float64(Float64(a * t) / c_m) * -4.0);
	elseif (a <= 1.7e+68)
		tmp = Float64(fma(Float64(x * y), 9.0, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[a, -0.0045], N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[a, 1.7e+68], N[(N[(N[(x * y), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < -0.00449999999999999966

   1. Initial program 79.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   if -0.00449999999999999966 < a < 1.70000000000000008e68

   1. Initial program 76.9%

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

   3. Taylor expanded in 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 70.8

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

   5. Applied rewrites 70.8%

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

   if 1.70000000000000008e68 < a

   1. Initial program 66.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 59.6

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

   5. Applied rewrites 59.6%

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

   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 60.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{a \cdot t}{c\_m} \cdot -4\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 (<= a -0.0045)
    (* (/ (* a t) c_m) -4.0)
    (if (<= a 1.7e+68)
      (/ (fma (* x 9.0) y b) (* z c_m))
      (/ (* -4.0 (* a t)) c_m)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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 (a <= -0.0045) {
		tmp = ((a * t) / c_m) * -4.0;
	} else if (a <= 1.7e+68) {
		tmp = fma((x * 9.0), y, b) / (z * c_m);
	} else {
		tmp = (-4.0 * (a * t)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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 (a <= -0.0045)
		tmp = Float64(Float64(Float64(a * t) / c_m) * -4.0);
	elseif (a <= 1.7e+68)
		tmp = Float64(fma(Float64(x * 9.0), y, b) / Float64(z * c_m));
	else
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c_m);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[a, -0.0045], N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[a, 1.7e+68], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < -0.00449999999999999966

   1. Initial program 79.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   if -0.00449999999999999966 < a < 1.70000000000000008e68

   1. Initial program 76.9%

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

   3. Taylor expanded in 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 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 70.8%

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

   if 1.70000000000000008e68 < a

   1. Initial program 66.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 59.6

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

   5. Applied rewrites 59.6%

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

   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 60.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c\_m}{b} \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 -6.5e+102)
    (/ b (* z c_m))
    (if (<= b 5e+15) (/ (* -4.0 (* a t)) c_m) (/ 1.0 (* (/ c_m b) z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6.5e+102) {
		tmp = b / (z * c_m);
	} else if (b <= 5e+15) {
		tmp = (-4.0 * (a * t)) / c_m;
	} else {
		tmp = 1.0 / ((c_m / b) * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-6.5d+102)) then
        tmp = b / (z * c_m)
    else if (b <= 5d+15) then
        tmp = ((-4.0d0) * (a * t)) / c_m
    else
        tmp = 1.0d0 / ((c_m / b) * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6.5e+102) {
		tmp = b / (z * c_m);
	} else if (b <= 5e+15) {
		tmp = (-4.0 * (a * t)) / c_m;
	} else {
		tmp = 1.0 / ((c_m / b) * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -6.5e+102:
		tmp = b / (z * c_m)
	elif b <= 5e+15:
		tmp = (-4.0 * (a * t)) / c_m
	else:
		tmp = 1.0 / ((c_m / b) * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -6.5e+102)
		tmp = Float64(b / Float64(z * c_m));
	elseif (b <= 5e+15)
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c_m);
	else
		tmp = Float64(1.0 / Float64(Float64(c_m / b) * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -6.5e+102)
		tmp = b / (z * c_m);
	elseif (b <= 5e+15)
		tmp = (-4.0 * (a * t)) / c_m;
	else
		tmp = 1.0 / ((c_m / b) * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -6.5e+102], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+15], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(1.0 / N[(N[(c$95$m / b), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -6.5000000000000004e102

   1. Initial program 76.3%

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

   3. 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 57.1

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

   5. Applied rewrites 57.1%

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

   if -6.5000000000000004e102 < b < 5e15

   1. Initial program 74.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   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 55.7%

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

   if 5e15 < b

   1. Initial program 77.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 62.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b} \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{b} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 -6.5e+102)
    (/ b (* z c_m))
    (if (<= b 5e+15) (/ (* -4.0 (* a t)) c_m) (/ (/ b c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6.5e+102) {
		tmp = b / (z * c_m);
	} else if (b <= 5e+15) {
		tmp = (-4.0 * (a * t)) / c_m;
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-6.5d+102)) then
        tmp = b / (z * c_m)
    else if (b <= 5d+15) then
        tmp = ((-4.0d0) * (a * t)) / c_m
    else
        tmp = (b / c_m) / z
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6.5e+102) {
		tmp = b / (z * c_m);
	} else if (b <= 5e+15) {
		tmp = (-4.0 * (a * t)) / c_m;
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -6.5e+102:
		tmp = b / (z * c_m)
	elif b <= 5e+15:
		tmp = (-4.0 * (a * t)) / c_m
	else:
		tmp = (b / c_m) / z
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -6.5e+102)
		tmp = Float64(b / Float64(z * c_m));
	elseif (b <= 5e+15)
		tmp = Float64(Float64(-4.0 * Float64(a * t)) / c_m);
	else
		tmp = Float64(Float64(b / c_m) / z);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -6.5e+102)
		tmp = b / (z * c_m);
	elseif (b <= 5e+15)
		tmp = (-4.0 * (a * t)) / c_m;
	else
		tmp = (b / c_m) / z;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -6.5e+102], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+15], N[(N[(-4.0 * N[(a * t), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -6.5000000000000004e102

   1. Initial program 76.3%

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

   3. 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 57.1

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

   5. Applied rewrites 57.1%

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

   if -6.5000000000000004e102 < b < 5e15

   1. Initial program 74.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right) - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   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 55.7%

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

   if 5e15 < b

   1. Initial program 77.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 62.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot t\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\\\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot t}{c\_m} \cdot -4\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (/ (* a t) c_m) -4.0)))
   (* c_s (if (<= t -7.4e-16) t_1 (if (<= t 2.7e-96) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (t <= -7.4e-16) {
		tmp = t_1;
	} else if (t <= 2.7e-96) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a * t) / c_m) * (-4.0d0)
    if (t <= (-7.4d-16)) then
        tmp = t_1
    else if (t <= 2.7d-96) then
        tmp = b / (z * c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a * t) / c_m) * -4.0;
	double tmp;
	if (t <= -7.4e-16) {
		tmp = t_1;
	} else if (t <= 2.7e-96) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((a * t) / c_m) * -4.0
	tmp = 0
	if t <= -7.4e-16:
		tmp = t_1
	elif t <= 2.7e-96:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(a * t) / c_m) * -4.0)
	tmp = 0.0
	if (t <= -7.4e-16)
		tmp = t_1;
	elseif (t <= 2.7e-96)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((a * t) / c_m) * -4.0;
	tmp = 0.0;
	if (t <= -7.4e-16)
		tmp = t_1;
	elseif (t <= 2.7e-96)
		tmp = b / (z * c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -7.4e-16], t$95$1, If[LessEqual[t, 2.7e-96], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.3999999999999999e-16 or 2.7e-96 < t

   1. Initial program 71.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 -7.3999999999999999e-16 < t < 2.7e-96

   1. Initial program 81.8%

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

   3. 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 49.0

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

   5. Applied rewrites 49.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 32.3

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

5. Applied rewrites 32.3%

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

6. Final simplification 32.3%

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

Developer Target 1: 80.5% 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 2024251 
(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)))