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

Percentage Accurate: 80.5% → 91.1%

Time: 18.9s

Alternatives: 19

Speedup: 0.7×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% 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: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{c\_m}, \frac{t}{-0.25}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 3.1e-18)
    (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) z) c_m)
    (fma
     (/ a c_m)
     (/ t -0.25)
     (fma x (* 9.0 (/ y (* c_m z))) (/ b (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 3.1e-18) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / z) / c_m;
	} else {
		tmp = fma((a / c_m), (t / -0.25), fma(x, (9.0 * (y / (c_m * z))), (b / (c_m * z))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 3.1e-18)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / z) / c_m);
	else
		tmp = fma(Float64(a / c_m), Float64(t / -0.25), fma(x, Float64(9.0 * Float64(y / Float64(c_m * z))), Float64(b / Float64(c_m * z))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 3.1e-18], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a / c$95$m), $MachinePrecision] * N[(t / -0.25), $MachinePrecision] + N[(x * N[(9.0 * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 3.10000000000000007e-18

   1. Initial program 81.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.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 egg-rr 84.9%

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

   if 3.10000000000000007e-18 < c

   1. Initial program 71.2%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 91.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. *-commutative N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. /-lowering-/.f64 N/A

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 94.3

          \[\leadsto \mathsf{fma}\left(\frac{a}{c}, \frac{t}{-0.25}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c \cdot z}, \frac{b}{\color{blue}{c \cdot z}}\right)\right) \]

   7. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{c}, \frac{t}{-0.25}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c\_m}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c\_m}\right)}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c\_m \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c\_m}, x \cdot \frac{\frac{9 \cdot y}{z}}{c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -4e+38)
      (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))
      (if (<= t_1 5e-309)
        (/ (fma 9.0 (/ (* x y) c_m) (/ (fma a (* -4.0 (* t z)) b) c_m)) z)
        (if (<= t_1 INFINITY)
          (/ 1.0 (/ (* c_m z) (fma x (* 9.0 y) (fma a (* t (* z -4.0)) b))))
          (fma a (/ (* t -4.0) c_m) (* x (/ (/ (* 9.0 y) z) c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double tmp;
	if (t_1 <= -4e+38) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	} else if (t_1 <= 5e-309) {
		tmp = fma(9.0, ((x * y) / c_m), (fma(a, (-4.0 * (t * z)), b) / c_m)) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0 / ((c_m * z) / fma(x, (9.0 * y), fma(a, (t * (z * -4.0)), b)));
	} else {
		tmp = fma(a, ((t * -4.0) / c_m), (x * (((9.0 * y) / z) / c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -4e+38)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z));
	elseif (t_1 <= 5e-309)
		tmp = Float64(fma(9.0, Float64(Float64(x * y) / c_m), Float64(fma(a, Float64(-4.0 * Float64(t * z)), b) / c_m)) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(c_m * z) / fma(x, Float64(9.0 * y), fma(a, Float64(t * Float64(z * -4.0)), b))));
	else
		tmp = fma(a, Float64(Float64(t * -4.0) / c_m), Float64(x * Float64(Float64(Float64(9.0 * y) / z) / c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -4e+38], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-309], N[(N[(9.0 * N[(N[(x * y), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(N[(a * N[(-4.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.0 / N[(N[(c$95$m * z), $MachinePrecision] / N[(x * N[(9.0 * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(x * N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -3.99999999999999991e38

   1. Initial program 90.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 91.5

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

   4. Applied egg-rr 91.5%

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

   if -3.99999999999999991e38 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 4.9999999999999995e-309

   1. Initial program 73.7%

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

   3. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

   5. Simplified 98.4%

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

   if 4.9999999999999995e-309 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 90.4%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 91.4

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

   4. Applied egg-rr 91.4%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.5

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

   6. Applied egg-rr 90.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 76.6

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

   8. Simplified 76.6%

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 88.1

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

   10. Applied egg-rr 88.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c}\right)}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c}, x \cdot \frac{\frac{9 \cdot y}{z}}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c\_m \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c\_m}, x \cdot \frac{\frac{9 \cdot y}{z}}{c\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e+47)
      (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))
      (if (<= t_1 5e-309)
        (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) c_m) z)
        (if (<= t_1 INFINITY)
          (/ 1.0 (/ (* c_m z) (fma x (* 9.0 y) (fma a (* t (* z -4.0)) b))))
          (fma a (/ (* t -4.0) c_m) (* x (/ (/ (* 9.0 y) z) c_m)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double tmp;
	if (t_1 <= -2e+47) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	} else if (t_1 <= 5e-309) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0 / ((c_m * z) / fma(x, (9.0 * y), fma(a, (t * (z * -4.0)), b)));
	} else {
		tmp = fma(a, ((t * -4.0) / c_m), (x * (((9.0 * y) / z) / c_m)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e+47)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z));
	elseif (t_1 <= 5e-309)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(c_m * z) / fma(x, Float64(9.0 * y), fma(a, Float64(t * Float64(z * -4.0)), b))));
	else
		tmp = fma(a, Float64(Float64(t * -4.0) / c_m), Float64(x * Float64(Float64(Float64(9.0 * y) / z) / c_m)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+47], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-309], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.0 / N[(N[(c$95$m * z), $MachinePrecision] / N[(x * N[(9.0 * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(x * N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -2.0000000000000001e47

   1. Initial program 90.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   if -2.0000000000000001e47 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 4.9999999999999995e-309

   1. Initial program 74.7%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 98.2%

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

   if 4.9999999999999995e-309 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 90.4%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 91.4

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

   4. Applied egg-rr 91.4%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.5

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

   6. Applied egg-rr 90.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 76.6

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

   8. Simplified 76.6%

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 88.1

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

   10. Applied egg-rr 88.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c}, x \cdot \frac{\frac{9 \cdot y}{z}}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c\_m \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e+47)
      (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))
      (if (<= t_1 5e-309)
        (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) c_m) z)
        (if (<= t_1 INFINITY)
          (/ 1.0 (/ (* c_m z) (fma x (* 9.0 y) (fma a (* t (* z -4.0)) b))))
          (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double tmp;
	if (t_1 <= -2e+47) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	} else if (t_1 <= 5e-309) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0 / ((c_m * z) / fma(x, (9.0 * y), fma(a, (t * (z * -4.0)), b)));
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e+47)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z));
	elseif (t_1 <= 5e-309)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(c_m * z) / fma(x, Float64(9.0 * y), fma(a, Float64(t * Float64(z * -4.0)), b))));
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+47], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-309], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.0 / N[(N[(c$95$m * z), $MachinePrecision] / N[(x * N[(9.0 * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -2.0000000000000001e47

   1. Initial program 90.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   if -2.0000000000000001e47 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 4.9999999999999995e-309

   1. Initial program 74.7%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 98.2%

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

   if 4.9999999999999995e-309 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 90.4%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 91.4

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

   4. Applied egg-rr 91.4%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.5

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

   6. Applied egg-rr 90.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 80.5

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

   8. Simplified 80.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e+47)
      (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))
      (if (<= t_1 0.0)
        (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) c_m) z)
        (if (<= t_1 INFINITY)
          t_1
          (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double tmp;
	if (t_1 <= -2e+47) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	} else if (t_1 <= 0.0) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e+47)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+47], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -2.0000000000000001e47

   1. Initial program 90.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

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

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate-*l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   if -2.0000000000000001e47 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 74.7%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 98.2%

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

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

   1. Initial program 90.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 80.5

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

   8. Simplified 80.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e-313)
      (/ (fma (* x y) 9.0 (fma (* t a) (* z -4.0) b)) (* c_m z))
      (if (<= t_1 0.0)
        (* t (/ (fma -4.0 a (/ b (* t z))) c_m))
        (if (<= t_1 INFINITY)
          t_1
          (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = fma((x * y), 9.0, fma((t * a), (z * -4.0), b)) / (c_m * z);
	} else if (t_1 <= 0.0) {
		tmp = t * (fma(-4.0, a, (b / (t * z))) / c_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e-313)
		tmp = Float64(fma(Float64(x * y), 9.0, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z));
	elseif (t_1 <= 0.0)
		tmp = Float64(t * Float64(fma(-4.0, a, Float64(b / Float64(t * z))) / c_m));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-313], N[(N[(N[(x * y), $MachinePrecision] * 9.0 + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t * N[(N[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1.99999999998e-313

   1. Initial program 92.5%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

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

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

      16. *-lowering-*.f64 N/A

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

      17. metadata-eval 91.7

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

   4. Applied egg-rr 91.7%

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

   if -1.99999999998e-313 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 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 x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 48.7

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

   5. Simplified 48.7%

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

   6. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   9. Taylor expanded in c around 0

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

       1. /-lowering-/.f64 N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lowering-*.f64 99.9

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

   11. Simplified 99.9%

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

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

   1. Initial program 90.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 80.5

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

   8. Simplified 80.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot y, 9, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z)))
        (t_2 (/ (fma (* x y) 9.0 (fma (* t a) (* z -4.0) b)) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e-313)
      t_2
      (if (<= t_1 0.0)
        (* t (/ (fma -4.0 a (/ b (* t z))) c_m))
        (if (<= t_1 INFINITY)
          t_2
          (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double t_2 = fma((x * y), 9.0, fma((t * a), (z * -4.0), b)) / (c_m * z);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = t * (fma(-4.0, a, (b / (t * z))) / c_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	t_2 = Float64(fma(Float64(x * y), 9.0, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e-313)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(t * Float64(fma(-4.0, a, Float64(b / Float64(t * z))) / c_m));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] * 9.0 + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-313], t$95$2, If[LessEqual[t$95$1, 0.0], N[(t * N[(N[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1.99999999998e-313 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.5%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-sub0 N/A

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

      8. associate-+l- N/A

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

      9. neg-sub0 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

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

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

      16. *-lowering-*.f64 N/A

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

      17. metadata-eval 91.6

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

   4. Applied egg-rr 91.6%

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

   if -1.99999999998e-313 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 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 x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 48.7

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

   5. Simplified 48.7%

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

   6. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   9. Taylor expanded in c around 0

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

       1. /-lowering-/.f64 N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lowering-*.f64 99.9

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

   11. Simplified 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 80.5

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

   8. Simplified 80.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, 9, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z)))
        (t_2 (/ (fma (* x 9.0) y (fma (* t a) (* z -4.0) b)) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -2e-313)
      t_2
      (if (<= t_1 0.0)
        (* t (/ (fma -4.0 a (/ b (* t z))) c_m))
        (if (<= t_1 INFINITY)
          t_2
          (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double t_2 = fma((x * 9.0), y, fma((t * a), (z * -4.0), b)) / (c_m * z);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = t * (fma(-4.0, a, (b / (t * z))) / c_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	t_2 = Float64(fma(Float64(x * 9.0), y, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -2e-313)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(t * Float64(fma(-4.0, a, Float64(b / Float64(t * z))) / c_m));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-313], t$95$2, If[LessEqual[t$95$1, 0.0], N[(t * N[(N[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -1.99999999998e-313 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 91.5%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 91.1

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

   4. Applied egg-rr 91.1%

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

   if -1.99999999998e-313 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 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 x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 48.7

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

   5. Simplified 48.7%

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

   6. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 95.9

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

   8. Simplified 95.9%

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

   9. Taylor expanded in c around 0

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

       1. /-lowering-/.f64 N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lowering-*.f64 99.9

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

   11. Simplified 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 80.5

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

   8. Simplified 80.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(a, t \cdot 4, \mathsf{fma}\left(x, \frac{y \cdot -9}{z}, 0 - \frac{b}{z}\right)\right)}{0 - c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right) \cdot \frac{1}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1
         (/
          (fma a (* t 4.0) (fma x (/ (* y -9.0) z) (- 0.0 (/ b z))))
          (- 0.0 c_m))))
   (*
    c_s
    (if (<= z -3e-11)
      t_1
      (if (<= z 0.0001)
        (/ (* (fma x (* 9.0 y) (fma a (* t (* z -4.0)) b)) (/ 1.0 c_m)) z)
        t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(a, (t * 4.0), fma(x, ((y * -9.0) / z), (0.0 - (b / z)))) / (0.0 - c_m);
	double tmp;
	if (z <= -3e-11) {
		tmp = t_1;
	} else if (z <= 0.0001) {
		tmp = (fma(x, (9.0 * y), fma(a, (t * (z * -4.0)), b)) * (1.0 / c_m)) / z;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(a, Float64(t * 4.0), fma(x, Float64(Float64(y * -9.0) / z), Float64(0.0 - Float64(b / z)))) / Float64(0.0 - c_m))
	tmp = 0.0
	if (z <= -3e-11)
		tmp = t_1;
	elseif (z <= 0.0001)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(a, Float64(t * Float64(z * -4.0)), b)) * Float64(1.0 / c_m)) / z);
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(a * N[(t * 4.0), $MachinePrecision] + N[(x * N[(N[(y * -9.0), $MachinePrecision] / z), $MachinePrecision] + N[(0.0 - N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -3e-11], t$95$1, If[LessEqual[z, 0.0001], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3e-11 or 1.00000000000000005e-4 < z

   1. Initial program 64.3%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 83.2%

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

   6. Taylor expanded in c around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   8. Simplified 88.2%

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

   if -3e-11 < z < 1.00000000000000005e-4

   1. Initial program 95.7%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 92.5

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

   4. Applied egg-rr 92.5%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. div-inv N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(-4 \cdot z\right), b\right)\right) \cdot \frac{1}{c}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t \cdot 4, \mathsf{fma}\left(x, \frac{y \cdot -9}{z}, 0 - \frac{b}{z}\right)\right)}{0 - c}\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)\right) \cdot \frac{1}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t \cdot 4, \mathsf{fma}\left(x, \frac{y \cdot -9}{z}, 0 - \frac{b}{z}\right)\right)}{0 - c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 2.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{-4}{c\_m}, t, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 2.55e-17)
    (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) z) c_m)
    (fma
     (* a (/ -4.0 c_m))
     t
     (fma x (* 9.0 (/ y (* c_m z))) (/ b (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 2.55e-17) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / z) / c_m;
	} else {
		tmp = fma((a * (-4.0 / c_m)), t, fma(x, (9.0 * (y / (c_m * z))), (b / (c_m * z))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 2.55e-17)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / z) / c_m);
	else
		tmp = fma(Float64(a * Float64(-4.0 / c_m)), t, fma(x, Float64(9.0 * Float64(y / Float64(c_m * z))), Float64(b / Float64(c_m * z))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 2.55e-17], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(9.0 * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 2.5500000000000001e-17

   1. Initial program 81.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.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 egg-rr 84.9%

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

   if 2.5500000000000001e-17 < c

   1. Initial program 71.2%

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

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 91.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.2

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

   7. Applied egg-rr 94.2%

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

Alternative 11: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 7.2e-9)
    (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) z) c_m)
    (fma
     a
     (* t (/ -4.0 c_m))
     (fma x (/ (* 9.0 y) (* c_m z)) (/ b (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 7.2e-9) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / z) / c_m;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), fma(x, ((9.0 * y) / (c_m * z)), (b / (c_m * z))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 7.2e-9)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / z) / c_m);
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), fma(x, Float64(Float64(9.0 * y) / Float64(c_m * z)), Float64(b / Float64(c_m * z))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 7.2e-9], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(9.0 * y), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 7.2e-9

   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. Step-by-step derivation

      1. associate-/r* N/A

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

      2. /-lowering-/.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 egg-rr 84.9%

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

   if 7.2e-9 < c

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

   5. Simplified 91.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{a \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \frac{9 \cdot y}{\mathsf{fma}\left(z, c\_m, 0\right)}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{b}{z}}{c\_m}\\ \mathbf{elif}\;a \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c\_m \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* t (/ (* a -4.0) c_m))))
   (*
    c_s
    (if (<= a -3e+21)
      t_1
      (if (<= a 1.35e-205)
        (* x (/ (* 9.0 y) (fma z c_m 0.0)))
        (if (<= a 3.7e-132)
          (/ (/ b z) c_m)
          (if (<= a 1.48e-5) (* 9.0 (* x (/ y (* c_m z)))) t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = t * ((a * -4.0) / c_m);
	double tmp;
	if (a <= -3e+21) {
		tmp = t_1;
	} else if (a <= 1.35e-205) {
		tmp = x * ((9.0 * y) / fma(z, c_m, 0.0));
	} else if (a <= 3.7e-132) {
		tmp = (b / z) / c_m;
	} else if (a <= 1.48e-5) {
		tmp = 9.0 * (x * (y / (c_m * z)));
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(t * Float64(Float64(a * -4.0) / c_m))
	tmp = 0.0
	if (a <= -3e+21)
		tmp = t_1;
	elseif (a <= 1.35e-205)
		tmp = Float64(x * Float64(Float64(9.0 * y) / fma(z, c_m, 0.0)));
	elseif (a <= 3.7e-132)
		tmp = Float64(Float64(b / z) / c_m);
	elseif (a <= 1.48e-5)
		tmp = Float64(9.0 * Float64(x * Float64(y / Float64(c_m * z))));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(t * N[(N[(a * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -3e+21], t$95$1, If[LessEqual[a, 1.35e-205], N[(x * N[(N[(9.0 * y), $MachinePrecision] / N[(z * c$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.7e-132], N[(N[(b / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[a, 1.48e-5], N[(9.0 * N[(x * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if a < -3e21 or 1.4800000000000001e-5 < a

   1. Initial program 72.7%

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

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

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

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval 70.6

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

   4. Applied egg-rr 70.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 72.9

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

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 67.0

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

   9. Simplified 67.0%

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

   if -3e21 < a < 1.35e-205

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 52.1

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

   5. Simplified 52.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. /-lowering-/.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lft-identity N/A

          \[\leadsto \frac{y \cdot 9}{\color{blue}{0 + c \cdot z}} \cdot x \]

      17. +-commutative N/A

          \[\leadsto \frac{y \cdot 9}{\color{blue}{c \cdot z + 0}} \cdot x \]

      18. *-commutative N/A

          \[\leadsto \frac{y \cdot 9}{\color{blue}{z \cdot c} + 0} \cdot x \]

      19. accelerator-lowering-fma.f64 54.2

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

   7. Applied egg-rr 54.2%

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

   if 1.35e-205 < a < 3.7000000000000002e-132

   1. Initial program 88.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   4. Step-by-step derivation

   5. Simplified 60.5%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]

      3. /-lowering-/.f64 60.7

         \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]

   7. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]

   if 3.7000000000000002e-132 < a < 1.4800000000000001e-5

   1. Initial program 87.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

      2. *-lowering-*.f64 57.5

         \[\leadsto \frac{9 \cdot \color{blue}{\left(x \cdot y\right)}}{z \cdot c} \]

   5. Simplified 57.5%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z \cdot c} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c \cdot z}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(9 \cdot x\right) \cdot \frac{y}{c \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot 9\right)} \cdot \frac{y}{c \cdot z} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right) \cdot x} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c \cdot z} \cdot x\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 9 \cdot \left(\color{blue}{\frac{y}{c \cdot z}} \cdot x\right) \]

      11. +-lft-identity N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{0 + c \cdot z}} \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{c \cdot z + 0}} \cdot x\right) \]

      13. *-commutative N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c} + 0} \cdot x\right) \]

      14. accelerator-lowering-fma.f64 53.8

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{\mathsf{fma}\left(z, c, 0\right)}} \cdot x\right) \]

   7. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{\mathsf{fma}\left(z, c, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]

      2. *-lowering-*.f64 53.8

         \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]

   9. Applied egg-rr 53.8%

      \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+21}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \frac{9 \cdot y}{\mathsf{fma}\left(z, c, 0\right)}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;a \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := 9 \cdot \left(x \cdot \frac{y}{c\_m \cdot z}\right)\\ t_2 := t \cdot \frac{a \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -130000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{b}{z}}{c\_m}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* 9.0 (* x (/ y (* c_m z))))) (t_2 (* t (/ (* a -4.0) c_m))))
   (*
    c_s
    (if (<= a -130000000.0)
      t_2
      (if (<= a 1.15e-205)
        t_1
        (if (<= a 4.8e-133) (/ (/ b z) c_m) (if (<= a 4.5e-10) t_1 t_2)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (c_m * z)));
	double t_2 = t * ((a * -4.0) / c_m);
	double tmp;
	if (a <= -130000000.0) {
		tmp = t_2;
	} else if (a <= 1.15e-205) {
		tmp = t_1;
	} else if (a <= 4.8e-133) {
		tmp = (b / z) / c_m;
	} else if (a <= 4.5e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 9.0d0 * (x * (y / (c_m * z)))
    t_2 = t * ((a * (-4.0d0)) / c_m)
    if (a <= (-130000000.0d0)) then
        tmp = t_2
    else if (a <= 1.15d-205) then
        tmp = t_1
    else if (a <= 4.8d-133) then
        tmp = (b / z) / c_m
    else if (a <= 4.5d-10) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (c_m * z)));
	double t_2 = t * ((a * -4.0) / c_m);
	double tmp;
	if (a <= -130000000.0) {
		tmp = t_2;
	} else if (a <= 1.15e-205) {
		tmp = t_1;
	} else if (a <= 4.8e-133) {
		tmp = (b / z) / c_m;
	} else if (a <= 4.5e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = 9.0 * (x * (y / (c_m * z)))
	t_2 = t * ((a * -4.0) / c_m)
	tmp = 0
	if a <= -130000000.0:
		tmp = t_2
	elif a <= 1.15e-205:
		tmp = t_1
	elif a <= 4.8e-133:
		tmp = (b / z) / c_m
	elif a <= 4.5e-10:
		tmp = t_1
	else:
		tmp = t_2
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(9.0 * Float64(x * Float64(y / Float64(c_m * z))))
	t_2 = Float64(t * Float64(Float64(a * -4.0) / c_m))
	tmp = 0.0
	if (a <= -130000000.0)
		tmp = t_2;
	elseif (a <= 1.15e-205)
		tmp = t_1;
	elseif (a <= 4.8e-133)
		tmp = Float64(Float64(b / z) / c_m);
	elseif (a <= 4.5e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = 9.0 * (x * (y / (c_m * z)));
	t_2 = t * ((a * -4.0) / c_m);
	tmp = 0.0;
	if (a <= -130000000.0)
		tmp = t_2;
	elseif (a <= 1.15e-205)
		tmp = t_1;
	elseif (a <= 4.8e-133)
		tmp = (b / z) / c_m;
	elseif (a <= 4.5e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(9.0 * N[(x * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[(a * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -130000000.0], t$95$2, If[LessEqual[a, 1.15e-205], t$95$1, If[LessEqual[a, 4.8e-133], N[(N[(b / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[a, 4.5e-10], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3e8 or 4.5e-10 < a

   1. Initial program 73.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot 9}, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      6. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + b\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(t \cdot a\right) \cdot \left(z \cdot 4\right)}\right)\right) + b\right)}{z \cdot c} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(t \cdot a\right) \cdot \left(\mathsf{neg}\left(z \cdot 4\right)\right)} + b\right)}{z \cdot c} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(t \cdot a, \mathsf{neg}\left(z \cdot 4\right), b\right)}\right)}{z \cdot c} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(\color{blue}{t \cdot a}, \mathsf{neg}\left(z \cdot 4\right), b\right)\right)}{z \cdot c} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      15. metadata-eval 71.1

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot \color{blue}{-4}, b\right)\right)}{z \cdot c} \]

   4. Applied egg-rr 71.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}{z \cdot c} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{z \cdot c}{\color{blue}{x \cdot \left(9 \cdot y\right)} + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z \cdot c}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, \color{blue}{9 \cdot y}, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(a \cdot t\right)} \cdot \left(z \cdot -4\right) + b\right)}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{a \cdot \left(t \cdot \left(z \cdot -4\right)\right)} + b\right)}} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, \color{blue}{t \cdot \left(z \cdot -4\right)}, b\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

      14. *-lowering-*.f64 73.3

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(-4 \cdot z\right), b\right)\right)}}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot t\right) \cdot -4}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot a\right)} \cdot -4}{c} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(a \cdot -4\right)}}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(-4 \cdot a\right)}}{c} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

      7. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\left(-4 \cdot \frac{a}{c}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      11. *-lowering-*.f64 66.0

          \[\leadsto t \cdot \frac{\color{blue}{-4 \cdot a}}{c} \]

   9. Simplified 66.0%

      \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

   if -1.3e8 < a < 1.15e-205 or 4.8e-133 < a < 4.5e-10

   1. Initial program 83.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]

      2. *-lowering-*.f64 53.2

         \[\leadsto \frac{9 \cdot \color{blue}{\left(x \cdot y\right)}}{z \cdot c} \]

   5. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z \cdot c} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(9 \cdot x\right) \cdot y}{\color{blue}{c \cdot z}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(9 \cdot x\right) \cdot \frac{y}{c \cdot z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot 9\right)} \cdot \frac{y}{c \cdot z} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(9 \cdot \frac{y}{c \cdot z}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right) \cdot x} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c \cdot z} \cdot x\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto 9 \cdot \left(\color{blue}{\frac{y}{c \cdot z}} \cdot x\right) \]

      11. +-lft-identity N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{0 + c \cdot z}} \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{c \cdot z + 0}} \cdot x\right) \]

      13. *-commutative N/A

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c} + 0} \cdot x\right) \]

      14. accelerator-lowering-fma.f64 54.2

          \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{\mathsf{fma}\left(z, c, 0\right)}} \cdot x\right) \]

   7. Applied egg-rr 54.2%

      \[\leadsto \color{blue}{9 \cdot \left(\frac{y}{\mathsf{fma}\left(z, c, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]

      2. *-lowering-*.f64 54.2

         \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]

   9. Applied egg-rr 54.2%

      \[\leadsto 9 \cdot \left(\frac{y}{\color{blue}{z \cdot c}} \cdot x\right) \]

   if 1.15e-205 < a < 4.8e-133

   1. Initial program 88.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   4. Step-by-step derivation

   5. Simplified 60.5%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]

      3. /-lowering-/.f64 60.7

         \[\leadsto \frac{\color{blue}{\frac{b}{z}}}{c} \]

   7. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\frac{\frac{b}{z}}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -130000000:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-205}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c\_m}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= a -5e+14)
    (* t (/ (fma -4.0 a (/ b (* t z))) c_m))
    (if (<= a 6.8e-7)
      (/ (fma 9.0 (* x y) b) (* c_m z))
      (fma a (* t (/ -4.0 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);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (a <= -5e+14) {
		tmp = t * (fma(-4.0, a, (b / (t * z))) / c_m);
	} else if (a <= 6.8e-7) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (a <= -5e+14)
		tmp = Float64(t * Float64(fma(-4.0, a, Float64(b / Float64(t * z))) / c_m));
	elseif (a <= 6.8e-7)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[a, -5e+14], N[(t * N[(N[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.8e-7], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < -5e14

   1. Initial program 70.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. *-lowering-*.f64 54.1

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Simplified 54.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c} + \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c} + \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \color{blue}{\frac{a}{c}}, \frac{b}{c \cdot \left(t \cdot z\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \color{blue}{\frac{b}{c \cdot \left(t \cdot z\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{\left(c \cdot t\right) \cdot z}}\right) \]

      6. *-commutative N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{z \cdot \left(c \cdot t\right)}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{z \cdot \left(c \cdot t\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \color{blue}{\left(t \cdot c\right)}}\right) \]

      9. *-lowering-*.f64 74.0

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \color{blue}{\left(t \cdot c\right)}}\right) \]

   8. Simplified 74.0%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \left(t \cdot c\right)}\right)} \]

   9. Taylor expanded in c around 0

      \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a + \frac{b}{t \cdot z}}{c}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a + \frac{b}{t \cdot z}}{c}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto t \cdot \frac{\color{blue}{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}}{c} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \frac{\mathsf{fma}\left(-4, a, \color{blue}{\frac{b}{t \cdot z}}\right)}{c} \]

       4. *-lowering-*.f64 72.4

          \[\leadsto t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{\color{blue}{t \cdot z}}\right)}{c} \]

   11. Simplified 72.4%

       \[\leadsto t \cdot \color{blue}{\frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}} \]

   if -5e14 < a < 6.79999999999999948e-7

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z \cdot c} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. *-lowering-*.f64 77.6

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Simplified 77.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

   if 6.79999999999999948e-7 < a

   1. Initial program 74.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. metadata-eval N/A

         \[\leadsto \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -4 \cdot \frac{t}{c}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t}{c} \cdot -4}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t \cdot -4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{t \cdot \frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{t \cdot \frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \color{blue}{\frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \frac{b}{c \cdot z}\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\left(x \cdot \frac{y}{c \cdot z}\right)} \cdot 9 + \frac{b}{c \cdot z}\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{x \cdot \left(\frac{y}{c \cdot z} \cdot 9\right)} + \frac{b}{c \cdot z}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} + \frac{b}{c \cdot z}\right) \]

   5. Simplified 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\frac{b}{c \cdot z}}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\frac{b}{c \cdot z}}\right) \]

      2. *-lowering-*.f64 80.1

         \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{\color{blue}{c \cdot z}}\right) \]

   8. Simplified 80.1%

      \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\frac{b}{c \cdot z}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -61000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* t (/ (fma -4.0 a (/ b (* t z))) c_m))))
   (*
    c_s
    (if (<= a -61000000000.0)
      t_1
      (if (<= a 1.85e-5) (/ (fma 9.0 (* x y) b) (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = t * (fma(-4.0, a, (b / (t * z))) / c_m);
	double tmp;
	if (a <= -61000000000.0) {
		tmp = t_1;
	} else if (a <= 1.85e-5) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(t * Float64(fma(-4.0, a, Float64(b / Float64(t * z))) / c_m))
	tmp = 0.0
	if (a <= -61000000000.0)
		tmp = t_1;
	elseif (a <= 1.85e-5)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(t * N[(N[(-4.0 * a + N[(b / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -61000000000.0], t$95$1, If[LessEqual[a, 1.85e-5], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.1e10 or 1.84999999999999991e-5 < a

   1. Initial program 73.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(t \cdot z\right)\right) \cdot -4} + b}{z \cdot c} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \left(\left(t \cdot z\right) \cdot -4\right)} + b}{z \cdot c} \]

      6. *-commutative N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(-4 \cdot \left(t \cdot z\right)\right)} + b}{z \cdot c} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right) \cdot -4}, b\right)}{z \cdot c} \]

      10. *-lowering-*.f64 59.3

          \[\leadsto \frac{\mathsf{fma}\left(a, \color{blue}{\left(t \cdot z\right)} \cdot -4, b\right)}{z \cdot c} \]

   5. Simplified 59.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \left(t \cdot z\right) \cdot -4, b\right)}}{z \cdot c} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c} + \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c} + \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{c \cdot \left(t \cdot z\right)}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \color{blue}{\frac{a}{c}}, \frac{b}{c \cdot \left(t \cdot z\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \color{blue}{\frac{b}{c \cdot \left(t \cdot z\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{\left(c \cdot t\right) \cdot z}}\right) \]

      6. *-commutative N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{z \cdot \left(c \cdot t\right)}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{\color{blue}{z \cdot \left(c \cdot t\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \color{blue}{\left(t \cdot c\right)}}\right) \]

      9. *-lowering-*.f64 75.9

         \[\leadsto t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \color{blue}{\left(t \cdot c\right)}}\right) \]

   8. Simplified 75.9%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{z \cdot \left(t \cdot c\right)}\right)} \]

   9. Taylor expanded in c around 0

      \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a + \frac{b}{t \cdot z}}{c}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a + \frac{b}{t \cdot z}}{c}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto t \cdot \frac{\color{blue}{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}}{c} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \frac{\mathsf{fma}\left(-4, a, \color{blue}{\frac{b}{t \cdot z}}\right)}{c} \]

       4. *-lowering-*.f64 78.4

          \[\leadsto t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{\color{blue}{t \cdot z}}\right)}{c} \]

   11. Simplified 78.4%

       \[\leadsto t \cdot \color{blue}{\frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}} \]

   if -6.1e10 < a < 1.84999999999999991e-5

   1. Initial program 83.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z \cdot c} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. *-lowering-*.f64 77.6

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Simplified 77.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -61000000000:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\mathsf{fma}\left(-4, a, \frac{b}{t \cdot z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (fma -4.0 (* t a) (/ b z)) c_m)))
   (*
    c_s
    (if (<= z -6.2e+16)
      t_1
      (if (<= z 4500000.0) (/ (fma 9.0 (* x y) b) (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(-4.0, (t * a), (b / z)) / c_m;
	double tmp;
	if (z <= -6.2e+16) {
		tmp = t_1;
	} else if (z <= 4500000.0) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c_m)
	tmp = 0.0
	if (z <= -6.2e+16)
		tmp = t_1;
	elseif (z <= 4500000.0)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -6.2e+16], t$95$1, If[LessEqual[z, 4500000.0], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2e16 or 4.5e6 < z

   1. Initial program 61.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x 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. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. metadata-eval N/A

         \[\leadsto \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{t}{c}\right)} \cdot -4 + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{t}{c} \cdot -4\right)} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      7. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(-4 \cdot \frac{t}{c}\right)} + \left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -4 \cdot \frac{t}{c}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t}{c} \cdot -4}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t \cdot -4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{t \cdot \frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{t \cdot \frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \color{blue}{\frac{-4}{c}}, 9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\frac{x \cdot y}{c \cdot z} \cdot 9} + \frac{b}{c \cdot z}\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{\left(x \cdot \frac{y}{c \cdot z}\right)} \cdot 9 + \frac{b}{c \cdot z}\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \color{blue}{x \cdot \left(\frac{y}{c \cdot z} \cdot 9\right)} + \frac{b}{c \cdot z}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, t \cdot \frac{-4}{c}, x \cdot \color{blue}{\left(9 \cdot \frac{y}{c \cdot z}\right)} + \frac{b}{c \cdot z}\right) \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-9 \cdot \frac{x \cdot y}{z} + \left(-1 \cdot \frac{b}{z} + 4 \cdot \left(a \cdot t\right)\right)}{c}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(-9 \cdot \frac{x \cdot y}{z} + \left(-1 \cdot \frac{b}{z} + 4 \cdot \left(a \cdot t\right)\right)\right)}{c}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(-9 \cdot \frac{x \cdot y}{z} + \left(-1 \cdot \frac{b}{z} + 4 \cdot \left(a \cdot t\right)\right)\right)}{c}} \]

   8. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(a, t \cdot 4, \mathsf{fma}\left(x, \frac{y \cdot -9}{z}, 0 - \frac{b}{z}\right)\right)}{c}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{b}{z} - 4 \cdot \left(a \cdot t\right)}}{c} \]
   10. Step-by-step derivation

       1. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{b}{z} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot t\right)}}{c} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{b}{z} + \color{blue}{-4} \cdot \left(a \cdot t\right)}{c} \]

       3. +-commutative N/A

          \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot t\right) + \frac{b}{z}}}{c} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}}{c} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{a \cdot t}, \frac{b}{z}\right)}{c} \]

       6. /-lowering-/.f64 71.3

          \[\leadsto \frac{\mathsf{fma}\left(-4, a \cdot t, \color{blue}{\frac{b}{z}}\right)}{c} \]

   11. Simplified 71.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}}{c} \]

   if -6.2e16 < z < 4.5e6

   1. Initial program 96.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. *-lowering-*.f64 83.9

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Simplified 83.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 4500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{a \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* t (/ (* a -4.0) c_m))))
   (*
    c_s
    (if (<= z -9.5e+53)
      t_1
      (if (<= z 8e+20) (/ (fma 9.0 (* x y) b) (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = t * ((a * -4.0) / c_m);
	double tmp;
	if (z <= -9.5e+53) {
		tmp = t_1;
	} else if (z <= 8e+20) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(t * Float64(Float64(a * -4.0) / c_m))
	tmp = 0.0
	if (z <= -9.5e+53)
		tmp = t_1;
	elseif (z <= 8e+20)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(t * N[(N[(a * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -9.5e+53], t$95$1, If[LessEqual[z, 8e+20], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000006e53 or 8e20 < z

   1. Initial program 60.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot 9}, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      6. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + b\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(t \cdot a\right) \cdot \left(z \cdot 4\right)}\right)\right) + b\right)}{z \cdot c} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(t \cdot a\right) \cdot \left(\mathsf{neg}\left(z \cdot 4\right)\right)} + b\right)}{z \cdot c} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(t \cdot a, \mathsf{neg}\left(z \cdot 4\right), b\right)}\right)}{z \cdot c} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(\color{blue}{t \cdot a}, \mathsf{neg}\left(z \cdot 4\right), b\right)\right)}{z \cdot c} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      15. metadata-eval 64.7

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot \color{blue}{-4}, b\right)\right)}{z \cdot c} \]

   4. Applied egg-rr 64.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}{z \cdot c} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{z \cdot c}{\color{blue}{x \cdot \left(9 \cdot y\right)} + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z \cdot c}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, \color{blue}{9 \cdot y}, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(a \cdot t\right)} \cdot \left(z \cdot -4\right) + b\right)}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{a \cdot \left(t \cdot \left(z \cdot -4\right)\right)} + b\right)}} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, \color{blue}{t \cdot \left(z \cdot -4\right)}, b\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

      14. *-lowering-*.f64 60.9

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

   6. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(-4 \cdot z\right), b\right)\right)}}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot t\right) \cdot -4}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot a\right)} \cdot -4}{c} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(a \cdot -4\right)}}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(-4 \cdot a\right)}}{c} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

      7. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\left(-4 \cdot \frac{a}{c}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      11. *-lowering-*.f64 68.3

          \[\leadsto t \cdot \frac{\color{blue}{-4 \cdot a}}{c} \]

   9. Simplified 68.3%

      \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

   if -9.5000000000000006e53 < z < 8e20

   1. Initial program 94.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{b + 9 \cdot \left(x \cdot y\right)}}{z \cdot c} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z \cdot c} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]

      3. *-lowering-*.f64 82.2

         \[\leadsto \frac{\mathsf{fma}\left(9, \color{blue}{x \cdot y}, b\right)}{z \cdot c} \]

   5. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, x \cdot y, b\right)}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+53}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := t \cdot \frac{a \cdot -4}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* t (/ (* a -4.0) c_m))))
   (* c_s (if (<= z -1.5e-99) t_1 (if (<= z 8.3e-119) (/ b (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = t * ((a * -4.0) / c_m);
	double tmp;
	if (z <= -1.5e-99) {
		tmp = t_1;
	} else if (z <= 8.3e-119) {
		tmp = b / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * ((a * (-4.0d0)) / c_m)
    if (z <= (-1.5d-99)) then
        tmp = t_1
    else if (z <= 8.3d-119) then
        tmp = b / (c_m * z)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = t * ((a * -4.0) / c_m);
	double tmp;
	if (z <= -1.5e-99) {
		tmp = t_1;
	} else if (z <= 8.3e-119) {
		tmp = b / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = t * ((a * -4.0) / c_m)
	tmp = 0
	if z <= -1.5e-99:
		tmp = t_1
	elif z <= 8.3e-119:
		tmp = b / (c_m * z)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(t * Float64(Float64(a * -4.0) / c_m))
	tmp = 0.0
	if (z <= -1.5e-99)
		tmp = t_1;
	elseif (z <= 8.3e-119)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = t * ((a * -4.0) / c_m);
	tmp = 0.0;
	if (z <= -1.5e-99)
		tmp = t_1;
	elseif (z <= 8.3e-119)
		tmp = b / (c_m * z);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(t * N[(N[(a * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.5e-99], t$95$1, If[LessEqual[z, 8.3e-119], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000003e-99 or 8.29999999999999946e-119 < z

   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. Step-by-step derivation

      1. associate-+l- N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}}{z \cdot c} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot 9}, y, \mathsf{neg}\left(\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)\right)}{z \cdot c} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]

      6. associate-+l- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + b\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \left(\mathsf{neg}\left(\color{blue}{\left(t \cdot a\right) \cdot \left(z \cdot 4\right)}\right)\right) + b\right)}{z \cdot c} \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\left(t \cdot a\right) \cdot \left(\mathsf{neg}\left(z \cdot 4\right)\right)} + b\right)}{z \cdot c} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \color{blue}{\mathsf{fma}\left(t \cdot a, \mathsf{neg}\left(z \cdot 4\right), b\right)}\right)}{z \cdot c} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(\color{blue}{t \cdot a}, \mathsf{neg}\left(z \cdot 4\right), b\right)\right)}{z \cdot c} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, \color{blue}{z \cdot \left(\mathsf{neg}\left(4\right)\right)}, b\right)\right)}{z \cdot c} \]

      15. metadata-eval 73.5

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot \color{blue}{-4}, b\right)\right)}{z \cdot c} \]

   4. Applied egg-rr 73.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}{z \cdot c} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{\left(x \cdot 9\right) \cdot y + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{z \cdot c}{\color{blue}{x \cdot \left(9 \cdot y\right)} + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z \cdot c}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{c \cdot z}}{x \cdot \left(9 \cdot y\right) + \left(\left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, \color{blue}{9 \cdot y}, \left(t \cdot a\right) \cdot \left(z \cdot -4\right) + b\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(a \cdot t\right)} \cdot \left(z \cdot -4\right) + b\right)}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{a \cdot \left(t \cdot \left(z \cdot -4\right)\right)} + b\right)}} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\mathsf{fma}\left(a, t \cdot \left(z \cdot -4\right), b\right)}\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, \color{blue}{t \cdot \left(z \cdot -4\right)}, b\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

      14. *-lowering-*.f64 71.5

          \[\leadsto \frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \color{blue}{\left(-4 \cdot z\right)}, b\right)\right)}} \]

   6. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{c \cdot z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, t \cdot \left(-4 \cdot z\right), b\right)\right)}}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(a \cdot t\right) \cdot -4}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot a\right)} \cdot -4}{c} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(a \cdot -4\right)}}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(-4 \cdot a\right)}}{c} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

      7. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\left(-4 \cdot \frac{a}{c}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-4 \cdot \frac{a}{c}\right)} \]

      9. associate-*r/ N/A

         \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto t \cdot \color{blue}{\frac{-4 \cdot a}{c}} \]

      11. *-lowering-*.f64 60.2

          \[\leadsto t \cdot \frac{\color{blue}{-4 \cdot a}}{c} \]

   9. Simplified 60.2%

      \[\leadsto \color{blue}{t \cdot \frac{-4 \cdot a}{c}} \]

   if -1.50000000000000003e-99 < z < 8.29999999999999946e-119

   1. Initial program 96.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   4. Step-by-step derivation

   5. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{c\_m \cdot z} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* c_m z))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    code = c_s * (b / (c_m * z))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (c_m * z))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(c_m * z)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (c_m * z));
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
4. Step-by-step derivation

5. Simplified 34.9%

   \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]

6. Final simplification 34.9%

   \[\leadsto \frac{b}{c \cdot z} \]
7. Add Preprocessing

Developer Target 1: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024194 
(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)))