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

Percentage Accurate: 79.4% → 90.5%

Time: 18.8s

Alternatives: 22

Speedup: 0.7×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% 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: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.15 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(t, a \cdot \frac{-4}{c\_m}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)}}\\ \end{array} \end{array} \]

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 1.15e-31)
    (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) z) c_m)
    (/
     1.0
     (/
      1.0
      (fma
       t
       (* a (/ -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 <= 1.15e-31) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / z) / c_m;
	} else {
		tmp = 1.0 / (1.0 / fma(t, (a * (-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 <= 1.15e-31)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / z) / c_m);
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(t, Float64(a * Float64(-4.0 / c_m)), 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, 1.15e-31], 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[(1.0 / N[(1.0 / N[(t * N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $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]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.1499999999999999e-31

   1. Initial program 82.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   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 83.4%

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

   if 1.1499999999999999e-31 < c

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   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 84.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

   7. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(t, a \cdot \frac{-4}{c}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -6.5 \cdot 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -6.5e+220)
      t_1
      (if (<= t_2 -1e-110)
        (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))
        (if (<= t_2 2e-265)
          (fma -4.0 (* a (/ t c_m)) (/ b (* c_m z)))
          (if (<= t_2 2e+224)
            (/ (fma (* x 9.0) y (fma (* t a) (* z -4.0) b)) (* c_m z))
            t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -6.5e+220) {
		tmp = t_1;
	} else if (t_2 <= -1e-110) {
		tmp = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	} else if (t_2 <= 2e-265) {
		tmp = fma(-4.0, (a * (t / c_m)), (b / (c_m * z)));
	} else if (t_2 <= 2e+224) {
		tmp = fma((x * 9.0), y, fma((t * a), (z * -4.0), b)) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -6.5e+220)
		tmp = t_1;
	elseif (t_2 <= -1e-110)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z));
	elseif (t_2 <= 2e-265)
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(b / Float64(c_m * z)));
	elseif (t_2 <= 2e+224)
		tmp = Float64(fma(Float64(x * 9.0), y, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -6.5e+220], t$95$1, If[LessEqual[t$95$2, -1e-110], 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$2, 2e-265], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+224], 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], t$95$1]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -6.5000000000000001e220 or 1.99999999999999994e224 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 51.6%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 58.7

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 58.7%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 81.4

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -6.5000000000000001e220 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.0000000000000001e-110

   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 92.4

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

   4. Applied egg-rr 92.4%

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

   if -1.0000000000000001e-110 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999997e-265

   1. Initial program 76.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.2

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

   8. Simplified 85.2%

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

   if 1.99999999999999997e-265 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999994e224

   1. Initial program 87.1%

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

      1. 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 89.3

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

   4. Applied egg-rr 89.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -6.5 \cdot 10^{+220}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot 9 \leq 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{elif}\;x \cdot 9 \leq 200000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 9}{c\_m} \cdot \frac{y}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (fma (* x 9.0) y (fma (* t a) (* z -4.0) b)) (* c_m z))))
   (*
    c_s
    (if (<= (* x 9.0) -5e+273)
      (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))
      (if (<= (* x 9.0) -2e-125)
        t_1
        (if (<= (* x 9.0) 1e-230)
          (fma -4.0 (* a (/ t c_m)) (/ b (* c_m z)))
          (if (<= (* x 9.0) 200000000.0)
            t_1
            (* (/ (* x 9.0) c_m) (/ y z)))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((x * 9.0), y, fma((t * a), (z * -4.0), b)) / (c_m * z);
	double tmp;
	if ((x * 9.0) <= -5e+273) {
		tmp = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	} else if ((x * 9.0) <= -2e-125) {
		tmp = t_1;
	} else if ((x * 9.0) <= 1e-230) {
		tmp = fma(-4.0, (a * (t / c_m)), (b / (c_m * z)));
	} else if ((x * 9.0) <= 200000000.0) {
		tmp = t_1;
	} else {
		tmp = ((x * 9.0) / c_m) * (y / z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(Float64(x * 9.0), y, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z))
	tmp = 0.0
	if (Float64(x * 9.0) <= -5e+273)
		tmp = Float64(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z));
	elseif (Float64(x * 9.0) <= -2e-125)
		tmp = t_1;
	elseif (Float64(x * 9.0) <= 1e-230)
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(b / Float64(c_m * z)));
	elseif (Float64(x * 9.0) <= 200000000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x * 9.0) / c_m) * Float64(y / z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(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[N[(x * 9.0), $MachinePrecision], -5e+273], N[(N[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], -2e-125], t$95$1, If[LessEqual[N[(x * 9.0), $MachinePrecision], 1e-230], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], 200000000.0], t$95$1, N[(N[(N[(x * 9.0), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x #s(literal 9 binary64)) < -4.99999999999999961e273

   1. Initial program 31.7%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 45.9

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 45.9%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 99.6

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -4.99999999999999961e273 < (*.f64 x #s(literal 9 binary64)) < -2.00000000000000002e-125 or 1.00000000000000005e-230 < (*.f64 x #s(literal 9 binary64)) < 2e8

   1. Initial program 87.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 88.6

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

   4. Applied egg-rr 88.6%

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

   if -2.00000000000000002e-125 < (*.f64 x #s(literal 9 binary64)) < 1.00000000000000005e-230

   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. 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 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 78.5

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

   8. Simplified 78.5%

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

   if 2e8 < (*.f64 x #s(literal 9 binary64))

   1. Initial program 71.9%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 40.7

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 40.7%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)} \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}}{c}} \]

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot 9\right) \cdot \frac{y}{z}}{\color{blue}{c \cdot 1}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{\frac{y}{z}}{1}} \]

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

         \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{\frac{y}{z}}{1}} \]

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

          \[\leadsto \color{blue}{\frac{x \cdot 9}{c}} \cdot \frac{\frac{y}{z}}{1} \]

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

          \[\leadsto \frac{\color{blue}{x \cdot 9}}{c} \cdot \frac{\frac{y}{z}}{1} \]

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

          \[\leadsto \frac{x \cdot 9}{c} \cdot \color{blue}{\frac{\frac{y}{z}}{1}} \]

      13. /-lowering-/.f64 48.6

          \[\leadsto \frac{x \cdot 9}{c} \cdot \frac{\color{blue}{\frac{y}{z}}}{1} \]

   9. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{x \cdot 9}{c} \cdot \frac{\frac{y}{z}}{1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;x \cdot 9 \leq 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;x \cdot 9 \leq 200000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 9}{c} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e+84)
      t_1
      (if (<= t_2 2e-20)
        (fma a (* t (/ -4.0 c_m)) (/ b (* c_m z)))
        (if (<= t_2 2e+224) (/ (fma (* x 9.0) 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 = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e+84) {
		tmp = t_1;
	} else if (t_2 <= 2e-20) {
		tmp = fma(a, (t * (-4.0 / c_m)), (b / (c_m * z)));
	} else if (t_2 <= 2e+224) {
		tmp = fma((x * 9.0), 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(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e+84)
		tmp = t_1;
	elseif (t_2 <= 2e-20)
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(b / Float64(c_m * z)));
	elseif (t_2 <= 2e+224)
		tmp = Float64(fma(Float64(x * 9.0), 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[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+84], t$95$1, If[LessEqual[t$95$2, 2e-20], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+224], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e84 or 1.99999999999999994e224 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.5%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 61.3

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 61.3%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 76.5

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -5.0000000000000001e84 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999989e-20

   1. Initial program 82.2%

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

   3. Taylor expanded in 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 84.8%

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

   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. *-commutative N/A

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

      3. *-lowering-*.f64 80.2

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

   8. Simplified 80.2%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999994e224

   1. Initial program 87.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 81.0

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e+84)
      t_1
      (if (<= t_2 2e-20)
        (fma -4.0 (* a (/ t c_m)) (/ b (* c_m z)))
        (if (<= t_2 2e+224) (/ (fma (* x 9.0) 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 = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e+84) {
		tmp = t_1;
	} else if (t_2 <= 2e-20) {
		tmp = fma(-4.0, (a * (t / c_m)), (b / (c_m * z)));
	} else if (t_2 <= 2e+224) {
		tmp = fma((x * 9.0), 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(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e+84)
		tmp = t_1;
	elseif (t_2 <= 2e-20)
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(b / Float64(c_m * z)));
	elseif (t_2 <= 2e+224)
		tmp = Float64(fma(Float64(x * 9.0), 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[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+84], t$95$1, If[LessEqual[t$95$2, 2e-20], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+224], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e84 or 1.99999999999999994e224 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.5%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 61.3

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 61.3%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 76.5

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -5.0000000000000001e84 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999989e-20

   1. Initial program 82.2%

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

   3. Taylor expanded in 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 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 79.5

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

   8. Simplified 79.5%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999994e224

   1. Initial program 87.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 81.0

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e+84)
      t_1
      (if (<= t_2 2e-20)
        (/ (fma -4.0 (* t a) (/ b z)) c_m)
        (if (<= t_2 2e+224) (/ (fma (* x 9.0) 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 = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e+84) {
		tmp = t_1;
	} else if (t_2 <= 2e-20) {
		tmp = fma(-4.0, (t * a), (b / z)) / c_m;
	} else if (t_2 <= 2e+224) {
		tmp = fma((x * 9.0), 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(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e+84)
		tmp = t_1;
	elseif (t_2 <= 2e-20)
		tmp = Float64(fma(-4.0, Float64(t * a), Float64(b / z)) / c_m);
	elseif (t_2 <= 2e+224)
		tmp = Float64(fma(Float64(x * 9.0), 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[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e+84], t$95$1, If[LessEqual[t$95$2, 2e-20], N[(N[(-4.0 * N[(t * a), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+224], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e84 or 1.99999999999999994e224 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 64.5%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 61.3

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 61.3%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 76.5

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -5.0000000000000001e84 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999989e-20

   1. Initial program 82.2%

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

   3. Taylor expanded in 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 84.8%

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

   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. *-commutative N/A

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

      3. *-lowering-*.f64 80.2

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

   8. Simplified 80.2%

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

   9. Taylor expanded in c around 0

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

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

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

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

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

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

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

       4. /-lowering-/.f64 76.3

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

   11. Simplified 76.3%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999994e224

   1. Initial program 87.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 81.0

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{\frac{z}{t\_1}}\\ \mathbf{elif}\;x \cdot 9 \leq 4 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b))))
   (*
    c_s
    (if (<= (* x 9.0) -5e+273)
      (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))
      (if (<= (* x 9.0) -2e-107)
        (/ (/ 1.0 c_m) (/ z t_1))
        (if (<= (* x 9.0) 4e-284)
          (fma -4.0 (* a (/ t c_m)) (/ b (* c_m z)))
          (/ (/ t_1 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 = fma(x, (9.0 * y), fma((t * a), (z * -4.0), b));
	double tmp;
	if ((x * 9.0) <= -5e+273) {
		tmp = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	} else if ((x * 9.0) <= -2e-107) {
		tmp = (1.0 / c_m) / (z / t_1);
	} else if ((x * 9.0) <= 4e-284) {
		tmp = fma(-4.0, (a * (t / c_m)), (b / (c_m * z)));
	} else {
		tmp = (t_1 / 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 = fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b))
	tmp = 0.0
	if (Float64(x * 9.0) <= -5e+273)
		tmp = Float64(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z));
	elseif (Float64(x * 9.0) <= -2e-107)
		tmp = Float64(Float64(1.0 / c_m) / Float64(z / t_1));
	elseif (Float64(x * 9.0) <= 4e-284)
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(b / Float64(c_m * z)));
	else
		tmp = Float64(Float64(t_1 / 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[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[N[(x * 9.0), $MachinePrecision], -5e+273], N[(N[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], -2e-107], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], 4e-284], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x #s(literal 9 binary64)) < -4.99999999999999961e273

   1. Initial program 31.7%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 45.9

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 45.9%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 99.6

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -4.99999999999999961e273 < (*.f64 x #s(literal 9 binary64)) < -2e-107

   1. Initial program 86.0%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 87.4%

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

   if -2e-107 < (*.f64 x #s(literal 9 binary64)) < 4.00000000000000015e-284

   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 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 66.2

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

   5. Simplified 66.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 80.9

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

   8. Simplified 80.9%

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

   if 4.00000000000000015e-284 < (*.f64 x #s(literal 9 binary64))

   1. Initial program 77.1%

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

      1. associate-/l/ N/A

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

      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 85.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}\\ \mathbf{elif}\;x \cdot 9 \leq 4 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c\_m}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, t\_1\right)}{c\_m \cdot z}\\ \mathbf{elif}\;x \cdot 9 \leq 4 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{b}{c\_m \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, t\_1\right)}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma (* t a) (* z -4.0) b)))
   (*
    c_s
    (if (<= (* x 9.0) -5e+273)
      (* (* x (/ 1.0 c_m)) (/ (* 9.0 y) z))
      (if (<= (* x 9.0) -2e-125)
        (/ (fma (* x 9.0) y t_1) (* c_m z))
        (if (<= (* x 9.0) 4e-284)
          (fma -4.0 (* a (/ t c_m)) (/ b (* c_m z)))
          (/ (/ (fma x (* 9.0 y) t_1) 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 = fma((t * a), (z * -4.0), b);
	double tmp;
	if ((x * 9.0) <= -5e+273) {
		tmp = (x * (1.0 / c_m)) * ((9.0 * y) / z);
	} else if ((x * 9.0) <= -2e-125) {
		tmp = fma((x * 9.0), y, t_1) / (c_m * z);
	} else if ((x * 9.0) <= 4e-284) {
		tmp = fma(-4.0, (a * (t / c_m)), (b / (c_m * z)));
	} else {
		tmp = (fma(x, (9.0 * y), t_1) / 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 = fma(Float64(t * a), Float64(z * -4.0), b)
	tmp = 0.0
	if (Float64(x * 9.0) <= -5e+273)
		tmp = Float64(Float64(x * Float64(1.0 / c_m)) * Float64(Float64(9.0 * y) / z));
	elseif (Float64(x * 9.0) <= -2e-125)
		tmp = Float64(fma(Float64(x * 9.0), y, t_1) / Float64(c_m * z));
	elseif (Float64(x * 9.0) <= 4e-284)
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(b / Float64(c_m * z)));
	else
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), t_1) / 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[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]}, N[(c$95$s * If[LessEqual[N[(x * 9.0), $MachinePrecision], -5e+273], N[(N[(x * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], -2e-125], N[(N[(N[(x * 9.0), $MachinePrecision] * y + t$95$1), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 9.0), $MachinePrecision], 4e-284], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + t$95$1), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x #s(literal 9 binary64)) < -4.99999999999999961e273

   1. Initial program 31.7%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in x around inf

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

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

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]

      2. *-lowering-*.f64 45.9

         \[\leadsto \frac{\frac{1}{c}}{\frac{z}{9 \cdot \color{blue}{\left(x \cdot y\right)}}} \]

   7. Simplified 45.9%

      \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\color{blue}{9 \cdot \left(x \cdot y\right)}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\frac{1}{c} \cdot \frac{1}{\frac{z}{9 \cdot \left(x \cdot y\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(9 \cdot x\right) \cdot y}}{z} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{c} \cdot \frac{\color{blue}{x \cdot \left(9 \cdot y\right)}}{z} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{c} \cdot \color{blue}{\left(x \cdot \frac{9 \cdot y}{z}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

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

         \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right)} \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\color{blue}{\frac{1}{c}} \cdot x\right) \cdot \frac{9 \cdot y}{z} \]

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

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \color{blue}{\frac{9 \cdot y}{z}} \]

      12. *-lowering-*.f64 99.6

          \[\leadsto \left(\frac{1}{c} \cdot x\right) \cdot \frac{\color{blue}{9 \cdot y}}{z} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{c} \cdot x\right) \cdot \frac{9 \cdot y}{z}} \]

   if -4.99999999999999961e273 < (*.f64 x #s(literal 9 binary64)) < -2.00000000000000002e-125

   1. Initial program 86.9%

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

      1. 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 86.9

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

   4. Applied egg-rr 86.9%

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

   if -2.00000000000000002e-125 < (*.f64 x #s(literal 9 binary64)) < 4.00000000000000015e-284

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]
   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 66.8

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

   5. Simplified 66.8%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 83.3

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

   8. Simplified 83.3%

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

   if 4.00000000000000015e-284 < (*.f64 x #s(literal 9 binary64))

   1. Initial program 77.1%

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

      1. associate-/l/ N/A

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

      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 85.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 9 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;\left(x \cdot \frac{1}{c}\right) \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;x \cdot 9 \leq 4 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(-4, \frac{a}{c\_m}, \frac{b}{t \cdot \left(c\_m \cdot z\right)}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c\_m}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}\\ \end{array} \end{array} \]

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 (<= z -6e+22)
    (* t (fma -4.0 (/ a c_m) (/ b (* t (* c_m z)))))
    (if (<= z 1.3e-174)
      (/ (fma 9.0 (/ (* x y) c_m) (/ (fma a (* -4.0 (* t z)) b) c_m)) z)
      (/ (/ 1.0 c_m) (/ z (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b))))))))

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 (z <= -6e+22) {
		tmp = t * fma(-4.0, (a / c_m), (b / (t * (c_m * z))));
	} else if (z <= 1.3e-174) {
		tmp = fma(9.0, ((x * y) / c_m), (fma(a, (-4.0 * (t * z)), b) / c_m)) / z;
	} else {
		tmp = (1.0 / c_m) / (z / fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)));
	}
	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 (z <= -6e+22)
		tmp = Float64(t * fma(-4.0, Float64(a / c_m), Float64(b / Float64(t * Float64(c_m * z)))));
	elseif (z <= 1.3e-174)
		tmp = Float64(fma(9.0, Float64(Float64(x * y) / c_m), Float64(fma(a, Float64(-4.0 * Float64(t * z)), b) / c_m)) / z);
	else
		tmp = Float64(Float64(1.0 / c_m) / Float64(z / fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b))));
	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[z, -6e+22], N[(t * N[(-4.0 * N[(a / c$95$m), $MachinePrecision] + N[(b / N[(t * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e-174], 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], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(z / N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -6e22

   1. Initial program 61.1%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 56.4

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

   5. Simplified 56.4%

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

   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. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 73.1

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

   8. Simplified 73.1%

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

   if -6e22 < z < 1.3000000000000001e-174

   1. Initial program 94.5%

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

   3. Taylor expanded in 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 95.1%

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

   if 1.3000000000000001e-174 < 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. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. associate-+l- N/A

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

      9. sub-neg N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}} \]

      14. associate-+l- N/A

          \[\leadsto \frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}} \]

      15. neg-sub0 N/A

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

   4. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(-4, \frac{a}{c}, \frac{b}{t \cdot \left(c \cdot z\right)}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

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 1.35e-35)
    (/ (/ (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 <= 1.35e-35) {
		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 <= 1.35e-35)
		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, 1.35e-35], 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 < 1.3499999999999999e-35

   1. Initial program 82.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   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 83.4%

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

   if 1.3499999999999999e-35 < c

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   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 84.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{c\_m \cdot z}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -5e+105)
      (* y (* 9.0 (/ x (* c_m z))))
      (if (<= t_1 4e-118)
        (/ (fma (* a (* z -4.0)) t b) (* c_m z))
        (/ (fma (* x 9.0) y b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (x * 9.0);
	double tmp;
	if (t_1 <= -5e+105) {
		tmp = y * (9.0 * (x / (c_m * z)));
	} else if (t_1 <= 4e-118) {
		tmp = fma((a * (z * -4.0)), t, b) / (c_m * z);
	} else {
		tmp = fma((x * 9.0), y, b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+105)
		tmp = Float64(y * Float64(9.0 * Float64(x / Float64(c_m * z))));
	elseif (t_1 <= 4e-118)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, b) / Float64(c_m * z));
	else
		tmp = Float64(fma(Float64(x * 9.0), y, b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+105], N[(y * N[(9.0 * N[(x / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-118], N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.00000000000000046e105

   1. Initial program 68.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 66.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.1

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

   7. Simplified 70.1%

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

   if -5.00000000000000046e105 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99999999999999994e-118

   1. Initial program 82.4%

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

   3. Taylor expanded in 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 73.7

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

   5. Simplified 73.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if 3.99999999999999994e-118 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot -9, -b\right)}{c\_m \cdot \left(-z\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -4e-31)
      (/ (fma x (* y -9.0) (- b)) (* c_m (- z)))
      (if (<= t_1 4e-118)
        (/ (fma a (* -4.0 (* t z)) b) (* c_m z))
        (/ (fma (* x 9.0) y b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (x * 9.0);
	double tmp;
	if (t_1 <= -4e-31) {
		tmp = fma(x, (y * -9.0), -b) / (c_m * -z);
	} else if (t_1 <= 4e-118) {
		tmp = fma(a, (-4.0 * (t * z)), b) / (c_m * z);
	} else {
		tmp = fma((x * 9.0), y, b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_1 <= -4e-31)
		tmp = Float64(fma(x, Float64(y * -9.0), Float64(-b)) / Float64(c_m * Float64(-z)));
	elseif (t_1 <= 4e-118)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(t * z)), b) / Float64(c_m * z));
	else
		tmp = Float64(fma(Float64(x * 9.0), y, b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -4e-31], N[(N[(x * N[(y * -9.0), $MachinePrecision] + (-b)), $MachinePrecision] / N[(c$95$m * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-118], N[(N[(a * N[(-4.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4e-31

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   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 75.9%

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

   6. Taylor expanded in z around 0

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

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

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

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

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

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

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

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

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

      5. /-lowering-/.f64 64.4

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

   8. Simplified 64.4%

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

   9. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. neg-sub0 N/A

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

       6. associate-+l- N/A

          \[\leadsto \frac{\color{blue}{0 - \left(b - -9 \cdot \left(x \cdot y\right)\right)}}{\mathsf{neg}\left(c \cdot z\right)} \]

       7. sub-neg N/A

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

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

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

       9. metadata-eval N/A

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

       10. neg-sub0 N/A

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

       11. mul-1-neg N/A

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

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

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

   11. Simplified 66.0%

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

   if -4e-31 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.99999999999999994e-118

   1. Initial program 81.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 78.6

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

   5. Simplified 78.6%

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

   if 3.99999999999999994e-118 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -4 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot -9, -b\right)}{c \cdot \left(-z\right)}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot \frac{x}{c\_m \cdot z}\right)\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+65}:\\ \;\;\;\;b \cdot \frac{1}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* y (* 9.0 (/ x (* c_m z))))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -1e+36) t_1 (if (<= t_2 5e+65) (* 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 = y * (9.0 * (x / (c_m * z)));
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -1e+36) {
		tmp = t_1;
	} else if (t_2 <= 5e+65) {
		tmp = b * (1.0 / (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) :: t_2
    real(8) :: tmp
    t_1 = y * (9.0d0 * (x / (c_m * z)))
    t_2 = y * (x * 9.0d0)
    if (t_2 <= (-1d+36)) then
        tmp = t_1
    else if (t_2 <= 5d+65) then
        tmp = b * (1.0d0 / (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 = y * (9.0 * (x / (c_m * z)));
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -1e+36) {
		tmp = t_1;
	} else if (t_2 <= 5e+65) {
		tmp = b * (1.0 / (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 = y * (9.0 * (x / (c_m * z)))
	t_2 = y * (x * 9.0)
	tmp = 0
	if t_2 <= -1e+36:
		tmp = t_1
	elif t_2 <= 5e+65:
		tmp = 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(y * Float64(9.0 * Float64(x / Float64(c_m * z))))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -1e+36)
		tmp = t_1;
	elseif (t_2 <= 5e+65)
		tmp = Float64(b * Float64(1.0 / 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 = y * (9.0 * (x / (c_m * z)));
	t_2 = y * (x * 9.0);
	tmp = 0.0;
	if (t_2 <= -1e+36)
		tmp = t_1;
	elseif (t_2 <= 5e+65)
		tmp = b * (1.0 / (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[(y * N[(9.0 * N[(x / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -1e+36], t$95$1, If[LessEqual[t$95$2, 5e+65], N[(b * N[(1.0 / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000004e36 or 4.99999999999999973e65 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 71.5%

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

      1. associate-/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 73.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 66.3

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

   7. Simplified 66.3%

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

   if -1.00000000000000004e36 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999973e65

   1. Initial program 83.3%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 47.5%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{b}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{z \cdot c} \cdot b} \]

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

         \[\leadsto \color{blue}{\frac{1}{z \cdot c} \cdot b} \]

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

         \[\leadsto \color{blue}{\frac{1}{z \cdot c}} \cdot b \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot z}} \cdot b \]

      6. *-lowering-*.f64 48.6

         \[\leadsto \frac{1}{\color{blue}{c \cdot z}} \cdot b \]

   7. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{1}{c \cdot z} \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+65}:\\ \;\;\;\;b \cdot \frac{1}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(x, 9 \cdot y, b\right) \cdot \frac{1}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.8e+25)
      t_1
      (if (<= z 2.2e-69) (* (fma x (* 9.0 y) 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(-4.0, (t * a), (b / z)) / c_m;
	double tmp;
	if (z <= -1.8e+25) {
		tmp = t_1;
	} else if (z <= 2.2e-69) {
		tmp = fma(x, (9.0 * y), 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(-4.0, Float64(t * a), Float64(b / z)) / c_m)
	tmp = 0.0
	if (z <= -1.8e+25)
		tmp = t_1;
	elseif (z <= 2.2e-69)
		tmp = Float64(fma(x, Float64(9.0 * y), b) * Float64(1.0 / 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, -1.8e+25], t$95$1, If[LessEqual[z, 2.2e-69], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] * N[(1.0 / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.80000000000000008e25 or 2.2e-69 < z

   1. Initial program 62.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 82.7%

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

   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. *-commutative N/A

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

      3. *-lowering-*.f64 65.5

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

   8. Simplified 65.5%

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

   9. Taylor expanded in c around 0

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

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

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

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

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

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

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

       4. /-lowering-/.f64 67.6

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

   11. Simplified 67.6%

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

   if -1.80000000000000008e25 < z < 2.2e-69

   1. Initial program 93.5%

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

      1. div-inv N/A

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

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

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 83.9%

      \[\leadsto \mathsf{fma}\left(x, 9 \cdot y, \color{blue}{b}\right) \cdot \frac{1}{z \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(x, 9 \cdot y, b\right) \cdot \frac{1}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t \cdot a, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot -9, -b\right)}{c\_m \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1.5e+29)
    (* (/ a c_m) (* t -4.0))
    (if (<= z 1.55e+208)
      (/ (fma x (* y -9.0) (- b)) (* c_m (- z)))
      (* -4.0 (/ (* t a) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1.5e+29) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 1.55e+208) {
		tmp = fma(x, (y * -9.0), -b) / (c_m * -z);
	} else {
		tmp = -4.0 * ((t * a) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1.5e+29)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (z <= 1.55e+208)
		tmp = Float64(fma(x, Float64(y * -9.0), Float64(-b)) / Float64(c_m * Float64(-z)));
	else
		tmp = Float64(-4.0 * Float64(Float64(t * a) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1.5e+29], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e+208], N[(N[(x * N[(y * -9.0), $MachinePrecision] + (-b)), $MachinePrecision] / N[(c$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e29

   1. Initial program 60.3%

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

      1. 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 76.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.5e29 < z < 1.5499999999999999e208

   1. Initial program 90.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 77.8%

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

   6. Taylor expanded in z around 0

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

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

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

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

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

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

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

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

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

      5. /-lowering-/.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. neg-sub0 N/A

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

       6. associate-+l- N/A

          \[\leadsto \frac{\color{blue}{0 - \left(b - -9 \cdot \left(x \cdot y\right)\right)}}{\mathsf{neg}\left(c \cdot z\right)} \]

       7. sub-neg N/A

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

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

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

       9. metadata-eval N/A

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

       10. neg-sub0 N/A

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

       11. mul-1-neg N/A

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

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

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

   11. Simplified 75.8%

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

   if 1.5499999999999999e208 < z

   1. Initial program 29.6%

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

      1. 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 73.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 48.0

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

   7. Simplified 48.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot -9, -b\right)}{c \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (/ a c_m) (* t -4.0))))
   (*
    c_s
    (if (<= z -4.8e+27)
      t_1
      (if (<= z 5.5e+188) (/ (fma (* x 9.0) 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 = (a / c_m) * (t * -4.0);
	double tmp;
	if (z <= -4.8e+27) {
		tmp = t_1;
	} else if (z <= 5.5e+188) {
		tmp = fma((x * 9.0), 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(Float64(a / c_m) * Float64(t * -4.0))
	tmp = 0.0
	if (z <= -4.8e+27)
		tmp = t_1;
	elseif (z <= 5.5e+188)
		tmp = Float64(fma(Float64(x * 9.0), 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[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -4.8e+27], t$95$1, If[LessEqual[z, 5.5e+188], N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999995e27 or 5.50000000000000013e188 < z

   1. Initial program 51.4%

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

      1. associate-/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 75.1%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 57.4

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

   7. Simplified 57.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 61.1

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

   9. Applied egg-rr 61.1%

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

   if -4.79999999999999995e27 < z < 5.50000000000000013e188

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 76.0

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

   5. Simplified 76.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 76.1

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

   7. Applied egg-rr 76.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -6e+29)
    (* (/ a c_m) (* t -4.0))
    (if (<= z 3.95e+210)
      (/ (fma 9.0 (* x y) b) (* c_m z))
      (* -4.0 (/ (* t a) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -6e+29) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 3.95e+210) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = -4.0 * ((t * a) / c_m);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -6e+29)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (z <= 3.95e+210)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = Float64(-4.0 * Float64(Float64(t * a) / c_m));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -6e+29], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.95e+210], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.9999999999999998e29

   1. Initial program 60.3%

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

      1. 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 76.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -5.9999999999999998e29 < z < 3.95000000000000014e210

   1. Initial program 90.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 75.7

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

   5. Simplified 75.7%

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

   if 3.95000000000000014e210 < z

   1. Initial program 29.6%

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

      1. 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 73.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 48.0

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

   7. Simplified 48.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;b \cdot \frac{1}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c\_m}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -8.2e+24)
    (* (/ a c_m) (* t -4.0))
    (if (<= z 6.8e-71) (* b (/ 1.0 (* c_m z))) (* a (* -4.0 (/ t c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -8.2e+24) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 6.8e-71) {
		tmp = b * (1.0 / (c_m * z));
	} else {
		tmp = a * (-4.0 * (t / c_m));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-8.2d+24)) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (z <= 6.8d-71) then
        tmp = b * (1.0d0 / (c_m * z))
    else
        tmp = a * ((-4.0d0) * (t / c_m))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -8.2e+24) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 6.8e-71) {
		tmp = b * (1.0 / (c_m * z));
	} else {
		tmp = a * (-4.0 * (t / c_m));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -8.2e+24:
		tmp = (a / c_m) * (t * -4.0)
	elif z <= 6.8e-71:
		tmp = b * (1.0 / (c_m * z))
	else:
		tmp = a * (-4.0 * (t / c_m))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -8.2e+24)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (z <= 6.8e-71)
		tmp = Float64(b * Float64(1.0 / Float64(c_m * z)));
	else
		tmp = Float64(a * Float64(-4.0 * Float64(t / c_m)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -8.2e+24)
		tmp = (a / c_m) * (t * -4.0);
	elseif (z <= 6.8e-71)
		tmp = b * (1.0 / (c_m * z));
	else
		tmp = a * (-4.0 * (t / c_m));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -8.2e+24], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e-71], N[(b * N[(1.0 / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000002e24

   1. Initial program 60.3%

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

      1. 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 76.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -8.2000000000000002e24 < z < 6.80000000000000007e-71

   1. Initial program 93.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 47.7%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot c}{b}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{z \cdot c} \cdot b} \]

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

         \[\leadsto \color{blue}{\frac{1}{z \cdot c} \cdot b} \]

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

         \[\leadsto \color{blue}{\frac{1}{z \cdot c}} \cdot b \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot z}} \cdot b \]

      6. *-lowering-*.f64 49.8

         \[\leadsto \frac{1}{\color{blue}{c \cdot z}} \cdot b \]

   7. Applied egg-rr 49.8%

      \[\leadsto \color{blue}{\frac{1}{c \cdot z} \cdot b} \]

   if 6.80000000000000007e-71 < 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. 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 81.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 38.9

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

   7. Simplified 38.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 37.6

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

   9. Applied egg-rr 37.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;b \cdot \frac{1}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-73}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c\_m}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1e+27)
    (* (/ a c_m) (* t -4.0))
    (if (<= z 4e-73) (/ b (* c_m z)) (* a (* -4.0 (/ t c_m)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1e+27) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 4e-73) {
		tmp = b / (c_m * z);
	} else {
		tmp = a * (-4.0 * (t / c_m));
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-1d+27)) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (z <= 4d-73) then
        tmp = b / (c_m * z)
    else
        tmp = a * ((-4.0d0) * (t / c_m))
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1e+27) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 4e-73) {
		tmp = b / (c_m * z);
	} else {
		tmp = a * (-4.0 * (t / c_m));
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -1e+27:
		tmp = (a / c_m) * (t * -4.0)
	elif z <= 4e-73:
		tmp = b / (c_m * z)
	else:
		tmp = a * (-4.0 * (t / c_m))
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1e+27)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (z <= 4e-73)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = Float64(a * Float64(-4.0 * Float64(t / c_m)));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -1e+27)
		tmp = (a / c_m) * (t * -4.0);
	elseif (z <= 4e-73)
		tmp = b / (c_m * z);
	else
		tmp = a * (-4.0 * (t / c_m));
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1e+27], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-73], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1e27

   1. Initial program 60.3%

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

      1. 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 76.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1e27 < z < 3.99999999999999999e-73

   1. Initial program 93.4%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 48.1%

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

   if 3.99999999999999999e-73 < z

   1. Initial program 64.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 81.8%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 38.4

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

   7. Simplified 38.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 37.1

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

   9. Applied egg-rr 37.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-73}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-4 \cdot \frac{t}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c\_m} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1.75e+25)
    (* (/ a c_m) (* t -4.0))
    (if (<= z 4.5e-70) (/ b (* c_m z)) (* -4.0 (/ (* t a) c_m))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1.75e+25) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 4.5e-70) {
		tmp = b / (c_m * z);
	} else {
		tmp = -4.0 * ((t * a) / c_m);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (z <= (-1.75d+25)) then
        tmp = (a / c_m) * (t * (-4.0d0))
    else if (z <= 4.5d-70) then
        tmp = b / (c_m * z)
    else
        tmp = (-4.0d0) * ((t * a) / c_m)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1.75e+25) {
		tmp = (a / c_m) * (t * -4.0);
	} else if (z <= 4.5e-70) {
		tmp = b / (c_m * z);
	} else {
		tmp = -4.0 * ((t * a) / c_m);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -1.75e+25:
		tmp = (a / c_m) * (t * -4.0)
	elif z <= 4.5e-70:
		tmp = b / (c_m * z)
	else:
		tmp = -4.0 * ((t * a) / c_m)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1.75e+25)
		tmp = Float64(Float64(a / c_m) * Float64(t * -4.0));
	elseif (z <= 4.5e-70)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = Float64(-4.0 * Float64(Float64(t * a) / c_m));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -1.75e+25)
		tmp = (a / c_m) * (t * -4.0);
	elseif (z <= 4.5e-70)
		tmp = b / (c_m * z);
	else
		tmp = -4.0 * ((t * a) / c_m);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1.75e+25], N[(N[(a / c$95$m), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e-70], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e25

   1. Initial program 60.3%

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

      1. 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 76.6%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.75e25 < z < 4.50000000000000022e-70

   1. Initial program 93.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 47.4%

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

   if 4.50000000000000022e-70 < z

   1. Initial program 63.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 81.3%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 39.5

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

   7. Simplified 39.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{c} \cdot \left(t \cdot -4\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \frac{t \cdot a}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* -4.0 (/ (* t a) c_m))))
   (* c_s (if (<= z -1.45e+23) t_1 (if (<= z 2.7e-69) (/ b (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -4.0 * ((t * a) / c_m);
	double tmp;
	if (z <= -1.45e+23) {
		tmp = t_1;
	} else if (z <= 2.7e-69) {
		tmp = b / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * ((t * a) / c_m)
    if (z <= (-1.45d+23)) then
        tmp = t_1
    else if (z <= 2.7d-69) then
        tmp = b / (c_m * z)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -4.0 * ((t * a) / c_m);
	double tmp;
	if (z <= -1.45e+23) {
		tmp = t_1;
	} else if (z <= 2.7e-69) {
		tmp = b / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = -4.0 * ((t * a) / c_m)
	tmp = 0
	if z <= -1.45e+23:
		tmp = t_1
	elif z <= 2.7e-69:
		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(-4.0 * Float64(Float64(t * a) / c_m))
	tmp = 0.0
	if (z <= -1.45e+23)
		tmp = t_1;
	elseif (z <= 2.7e-69)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = -4.0 * ((t * a) / c_m);
	tmp = 0.0;
	if (z <= -1.45e+23)
		tmp = t_1;
	elseif (z <= 2.7e-69)
		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[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.45e+23], t$95$1, If[LessEqual[z, 2.7e-69], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45000000000000006e23 or 2.6999999999999997e-69 < z

   1. Initial program 62.2%

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

      1. 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 79.2%

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

   5. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 50.1

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

   7. Simplified 50.1%

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

   if -1.45000000000000006e23 < z < 2.6999999999999997e-69

   1. Initial program 93.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 47.4%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.5% accurate, 2.8× speedup

Math

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

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 78.7%

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

3. Taylor expanded in b around inf

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

5. Simplified 34.4%

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

6. Final simplification 34.4%

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

Developer Target 1: 80.7% 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 2024204 
(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)))