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

Percentage Accurate: 80.0% → 88.3%

Time: 18.8s

Alternatives: 19

Speedup: 0.7×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ t_2 := \frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c\_m}, \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z)))
        (t_2 (/ (fma (* a (* z -4.0)) t (fma x (* 9.0 y) b)) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -5e+146)
      t_2
      (if (<= t_1 1e-62)
        (/ (/ (fma x (* 9.0 y) (fma (* t a) (* z -4.0) b)) c_m) z)
        (if (<= t_1 INFINITY)
          t_2
          (* a (fma -4.0 (/ t c_m) (/ b (* a (* c_m z)))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double t_2 = fma((a * (z * -4.0)), t, fma(x, (9.0 * y), b)) / (c_m * z);
	double tmp;
	if (t_1 <= -5e+146) {
		tmp = t_2;
	} else if (t_1 <= 1e-62) {
		tmp = (fma(x, (9.0 * y), fma((t * a), (z * -4.0), b)) / c_m) / z;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = a * fma(-4.0, (t / c_m), (b / (a * (c_m * z))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	t_2 = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(9.0 * y), b)) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -5e+146)
		tmp = t_2;
	elseif (t_1 <= 1e-62)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), fma(Float64(t * a), Float64(z * -4.0), b)) / c_m) / z);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(a * fma(-4.0, Float64(t / c_m), Float64(b / Float64(a * Float64(c_m * z)))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(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]}, N[(c$95$s * If[LessEqual[t$95$1, -5e+146], t$95$2, If[LessEqual[t$95$1, 1e-62], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -4.9999999999999999e146 or 1e-62 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 85.8%

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

      1. 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 86.6

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

   4. Applied egg-rr 86.6%

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

   if -4.9999999999999999e146 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 1e-62

   1. Initial program 76.3%

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

      1. 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 97.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 5.2

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

   5. Simplified 5.2%

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

   6. Taylor expanded in a around inf

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 68.3

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

   8. Simplified 68.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c}}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c}, \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, 9 \cdot y, b\right)\\ t_2 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ t_3 := \frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, t\_1\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t\_1}{c\_m} \cdot \frac{1}{z}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c\_m}, \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (fma x (* 9.0 y) b))
        (t_2 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z)))
        (t_3 (/ (fma (* a (* z -4.0)) t t_1) (* c_m z))))
   (*
    c_s
    (if (<= t_2 -5e-88)
      t_3
      (if (<= t_2 0.0)
        (* (/ t_1 c_m) (/ 1.0 z))
        (if (<= t_2 INFINITY)
          t_3
          (* a (fma -4.0 (/ t c_m) (/ b (* a (* 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), b);
	double t_2 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double t_3 = fma((a * (z * -4.0)), t, t_1) / (c_m * z);
	double tmp;
	if (t_2 <= -5e-88) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = (t_1 / c_m) * (1.0 / z);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = a * fma(-4.0, (t / c_m), (b / (a * (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), b)
	t_2 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	t_3 = Float64(fma(Float64(a * Float64(z * -4.0)), t, t_1) / Float64(c_m * z))
	tmp = 0.0
	if (t_2 <= -5e-88)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t_1 / c_m) * Float64(1.0 / z));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(a * fma(-4.0, Float64(t / c_m), Float64(b / Float64(a * 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[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] * t + t$95$1), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e-88], t$95$3, If[LessEqual[t$95$2, 0.0], N[(N[(t$95$1 / c$95$m), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -5.00000000000000009e-88 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 87.8%

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

      1. 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 88.5

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

   4. Applied egg-rr 88.5%

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

   if -5.00000000000000009e-88 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 49.0%

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

      1. 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. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 78.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 5.2

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

   5. Simplified 5.2%

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

   6. Taylor expanded in a around inf

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 68.3

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

   8. Simplified 68.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c} \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c}, \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ t_2 := \frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m} \cdot \frac{1}{z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c\_m}, \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (+ b (- (* y (* x 9.0)) (* a (* t (* z 4.0))))) (* c_m z)))
        (t_2 (/ (fma (* x 9.0) y (fma (* t a) (* z -4.0) b)) (* c_m z))))
   (*
    c_s
    (if (<= t_1 -5e-88)
      t_2
      (if (<= t_1 0.0)
        (* (/ (fma x (* 9.0 y) b) c_m) (/ 1.0 z))
        (if (<= t_1 INFINITY)
          t_2
          (* a (fma -4.0 (/ t c_m) (/ b (* a (* c_m z)))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (b + ((y * (x * 9.0)) - (a * (t * (z * 4.0))))) / (c_m * z);
	double t_2 = fma((x * 9.0), y, fma((t * a), (z * -4.0), b)) / (c_m * z);
	double tmp;
	if (t_1 <= -5e-88) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (fma(x, (9.0 * y), b) / c_m) * (1.0 / z);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = a * fma(-4.0, (t / c_m), (b / (a * (c_m * z))));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(b + Float64(Float64(y * Float64(x * 9.0)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z))
	t_2 = Float64(fma(Float64(x * 9.0), y, fma(Float64(t * a), Float64(z * -4.0), b)) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= -5e-88)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), b) / c_m) * Float64(1.0 / z));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(a * fma(-4.0, Float64(t / c_m), Float64(b / Float64(a * Float64(c_m * z)))));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(b + N[(N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 9.0), $MachinePrecision] * y + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -5e-88], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -5.00000000000000009e-88 or 0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 87.8%

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

      1. associate-+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 85.6

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

   4. Applied egg-rr 85.6%

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

   if -5.00000000000000009e-88 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 0.0

   1. Initial program 49.0%

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

      1. 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. div-inv N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 78.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 5.2

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

   5. Simplified 5.2%

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

   6. Taylor expanded in a around inf

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 68.3

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

   8. Simplified 68.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c} \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot 9, y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c}, \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(9, x \cdot y, b\right)\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{t\_1}{z}}{c\_m}\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{t\_1}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 9.0 (* x y) b)) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e-50)
      (/ (/ t_1 z) c_m)
      (if (<= t_2 1e-21)
        (/ (fma (* z (* a -4.0)) t b) (* c_m z))
        (if (<= t_2 5e+186) (/ t_1 (* c_m z)) (/ (/ t_2 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(9.0, (x * y), b);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e-50) {
		tmp = (t_1 / z) / c_m;
	} else if (t_2 <= 1e-21) {
		tmp = fma((z * (a * -4.0)), t, b) / (c_m * z);
	} else if (t_2 <= 5e+186) {
		tmp = t_1 / (c_m * z);
	} else {
		tmp = (t_2 / 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(9.0, Float64(x * y), b)
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e-50)
		tmp = Float64(Float64(t_1 / z) / c_m);
	elseif (t_2 <= 1e-21)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, b) / Float64(c_m * z));
	elseif (t_2 <= 5e+186)
		tmp = Float64(t_1 / Float64(c_m * z));
	else
		tmp = Float64(Float64(t_2 / 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[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e-50], N[(N[(t$95$1 / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1e-21], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+186], N[(t$95$1 / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50

   1. Initial program 78.9%

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

      1. associate-/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 74.9%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-lowering-*.f64 72.6

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

   7. Simplified 72.6%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999954e186

   1. Initial program 86.4%

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

   3. Taylor expanded in 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 4.99999999999999954e186 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 9.0 (* x y) b) (* c_m z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e-50)
      t_1
      (if (<= t_2 1e-21)
        (/ (fma (* z (* a -4.0)) t b) (* c_m z))
        (if (<= t_2 5e+186) t_1 (/ (/ t_2 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(9.0, (x * y), b) / (c_m * z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e-50) {
		tmp = t_1;
	} else if (t_2 <= 1e-21) {
		tmp = fma((z * (a * -4.0)), t, b) / (c_m * z);
	} else if (t_2 <= 5e+186) {
		tmp = t_1;
	} else {
		tmp = (t_2 / 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(fma(9.0, Float64(x * y), b) / Float64(c_m * z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e-50)
		tmp = t_1;
	elseif (t_2 <= 1e-21)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, b) / Float64(c_m * z));
	elseif (t_2 <= 5e+186)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_2 / 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[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * 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-50], t$95$1, If[LessEqual[t$95$2, 1e-21], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+186], t$95$1, N[(N[(t$95$2 / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50 or 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999954e186

   1. Initial program 81.8%

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

   3. Taylor expanded in 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 72.9

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

   5. Simplified 72.9%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

   if 4.99999999999999954e186 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c\_m \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 9.0 (* x y) b) (* c_m z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e-50)
      t_1
      (if (<= t_2 1e-21)
        (/ (fma a (* -4.0 (* t z)) b) (* c_m z))
        (if (<= t_2 5e+186) t_1 (/ (/ t_2 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(9.0, (x * y), b) / (c_m * z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e-50) {
		tmp = t_1;
	} else if (t_2 <= 1e-21) {
		tmp = fma(a, (-4.0 * (t * z)), b) / (c_m * z);
	} else if (t_2 <= 5e+186) {
		tmp = t_1;
	} else {
		tmp = (t_2 / 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(fma(9.0, Float64(x * y), b) / Float64(c_m * z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e-50)
		tmp = t_1;
	elseif (t_2 <= 1e-21)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(t * z)), b) / Float64(c_m * z));
	elseif (t_2 <= 5e+186)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_2 / 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[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * 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-50], t$95$1, If[LessEqual[t$95$2, 1e-21], N[(N[(a * N[(-4.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+186], t$95$1, N[(N[(t$95$2 / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50 or 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999954e186

   1. Initial program 81.8%

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

   3. Taylor expanded in 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 72.9

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

   5. Simplified 72.9%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   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 68.2

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

   5. Simplified 68.2%

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

   if 4.99999999999999954e186 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(t \cdot z\right), b\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(x \cdot 9\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{a \cdot \mathsf{fma}\left(-4, t, \mathsf{fma}\left(x \cdot 9, \frac{y}{a \cdot z}, \frac{b}{a \cdot z}\right)\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (fma -4.0 t (fma (* x 9.0) (/ y (* a z)) (/ b (* a z)))))
          c_m)))
   (*
    c_s
    (if (<= z -2.2e+74)
      t_1
      (if (<= z 1.4e+44)
        (/ (fma (* a (* z -4.0)) t (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 * fma(-4.0, t, fma((x * 9.0), (y / (a * z)), (b / (a * z))))) / c_m;
	double tmp;
	if (z <= -2.2e+74) {
		tmp = t_1;
	} else if (z <= 1.4e+44) {
		tmp = fma((a * (z * -4.0)), t, 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 * fma(-4.0, t, fma(Float64(x * 9.0), Float64(y / Float64(a * z)), Float64(b / Float64(a * z))))) / c_m)
	tmp = 0.0
	if (z <= -2.2e+74)
		tmp = t_1;
	elseif (z <= 1.4e+44)
		tmp = Float64(fma(Float64(a * Float64(z * -4.0)), t, fma(x, Float64(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 * N[(-4.0 * t + N[(N[(x * 9.0), $MachinePrecision] * N[(y / N[(a * z), $MachinePrecision]), $MachinePrecision] + N[(b / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -2.2e+74], t$95$1, If[LessEqual[z, 1.4e+44], 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], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e74 or 1.4e44 < z

   1. Initial program 56.7%

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

      1. associate-/r* N/A

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

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

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

   4. Applied egg-rr 79.1%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 81.4

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

   7. Simplified 81.4%

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

   if -2.2000000000000001e74 < z < 1.4e44

   1. Initial program 91.7%

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

      1. 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 92.7

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

   4. Applied egg-rr 92.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-4, t, \mathsf{fma}\left(x \cdot 9, \frac{y}{a \cdot z}, \frac{b}{a \cdot z}\right)\right)}{c}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(z \cdot -4\right), t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \mathsf{fma}\left(-4, t, \mathsf{fma}\left(x \cdot 9, \frac{y}{a \cdot z}, \frac{b}{a \cdot z}\right)\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c\_m}, \frac{b}{a \cdot \left(c\_m \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -5e-50)
      (* (/ (fma x (* 9.0 y) b) z) (/ 1.0 c_m))
      (if (<= t_1 1e-21)
        (* a (fma -4.0 (/ t c_m) (/ b (* a (* c_m z)))))
        (/ (/ (fma y (* x 9.0) 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-50) {
		tmp = (fma(x, (9.0 * y), b) / z) * (1.0 / c_m);
	} else if (t_1 <= 1e-21) {
		tmp = a * fma(-4.0, (t / c_m), (b / (a * (c_m * z))));
	} else {
		tmp = (fma(y, (x * 9.0), 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-50)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), b) / z) * Float64(1.0 / c_m));
	elseif (t_1 <= 1e-21)
		tmp = Float64(a * fma(-4.0, Float64(t / c_m), Float64(b / Float64(a * Float64(c_m * z)))));
	else
		tmp = Float64(Float64(fma(y, Float64(x * 9.0), b) / 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-50], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(a * N[(-4.0 * N[(t / c$95$m), $MachinePrecision] + N[(b / N[(a * N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(x * 9.0), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50

   1. Initial program 78.9%

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

      1. associate-/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. div-inv N/A

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

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

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

   4. Applied egg-rr 75.0%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-lowering-*.f64 72.6

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

   7. Simplified 72.6%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

   6. Taylor expanded in a around inf

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 73.9

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

   8. Simplified 73.9%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 82.4

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

   7. Simplified 82.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(-4, \frac{t}{c}, \frac{b}{a \cdot \left(c \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.55 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c\_m}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

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 1.55e-41)
    (/ (/ (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 <= 1.55e-41) {
		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 <= 1.55e-41)
		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(Float64(t * -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, 1.55e-41], 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[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(x * N[(9.0 * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.55e-41

   1. Initial program 79.1%

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

      1. associate-/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 85.3%

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

   if 1.55e-41 < c

   1. Initial program 69.4%

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

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

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

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

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

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

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

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

      19. *-commutative N/A

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

   7. Simplified 89.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.55 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t \cdot -4}{c}, \mathsf{fma}\left(x, 9 \cdot \frac{y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \end{array} \end{array} \]

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 1.5e-46)
    (/ (/ (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 <= 1.5e-46) {
		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 <= 1.5e-46)
		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, 1.5e-46], 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 < 1.49999999999999994e-46

   1. Initial program 79.1%

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

      1. associate-/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 85.3%

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

   if 1.49999999999999994e-46 < c

   1. Initial program 69.4%

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

   3. Taylor expanded in 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 89.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t \cdot a, z \cdot -4, b\right)\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, a \cdot \left(z \cdot -4\right), b\right)}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -1e-125)
      (* (/ (fma x (* 9.0 y) b) z) (/ 1.0 c_m))
      (if (<= t_1 1e-21)
        (/ (/ (fma t (* a (* z -4.0)) b) c_m) z)
        (/ (/ (fma y (* x 9.0) 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 <= -1e-125) {
		tmp = (fma(x, (9.0 * y), b) / z) * (1.0 / c_m);
	} else if (t_1 <= 1e-21) {
		tmp = (fma(t, (a * (z * -4.0)), b) / c_m) / z;
	} else {
		tmp = (fma(y, (x * 9.0), 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 <= -1e-125)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), b) / z) * Float64(1.0 / c_m));
	elseif (t_1 <= 1e-21)
		tmp = Float64(Float64(fma(t, Float64(a * Float64(z * -4.0)), b) / c_m) / z);
	else
		tmp = Float64(Float64(fma(y, Float64(x * 9.0), b) / 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, -1e-125], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(N[(N[(t * N[(a * N[(z * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y * N[(x * 9.0), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000001e-125

   1. Initial program 79.1%

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

      1. associate-/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. div-inv N/A

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

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

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

   4. Applied egg-rr 78.0%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-lowering-*.f64 69.9

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

   7. Simplified 69.9%

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

   if -1.00000000000000001e-125 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 70.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 69.6

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

   5. Simplified 69.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 75.3

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

   7. Applied egg-rr 75.3%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 82.4

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

   7. Simplified 82.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, a \cdot \left(z \cdot -4\right), b\right)}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -5e-50)
      (* (/ (fma x (* 9.0 y) b) z) (/ 1.0 c_m))
      (if (<= t_1 1e-21)
        (/ (fma (* z (* a -4.0)) t b) (* c_m z))
        (/ (/ (fma y (* x 9.0) 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-50) {
		tmp = (fma(x, (9.0 * y), b) / z) * (1.0 / c_m);
	} else if (t_1 <= 1e-21) {
		tmp = fma((z * (a * -4.0)), t, b) / (c_m * z);
	} else {
		tmp = (fma(y, (x * 9.0), 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-50)
		tmp = Float64(Float64(fma(x, Float64(9.0 * y), b) / z) * Float64(1.0 / c_m));
	elseif (t_1 <= 1e-21)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(fma(y, Float64(x * 9.0), b) / 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-50], N[(N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(x * 9.0), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50

   1. Initial program 78.9%

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

      1. associate-/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. div-inv N/A

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

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

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

   4. Applied egg-rr 75.0%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-lowering-*.f64 72.6

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

   7. Simplified 72.6%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 82.4

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

   7. Simplified 82.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c\_m}}{z}\\ \end{array} \end{array} \end{array} \]

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 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_1 -5e-50)
      (/ (/ (fma 9.0 (* x y) b) z) c_m)
      (if (<= t_1 1e-21)
        (/ (fma (* z (* a -4.0)) t b) (* c_m z))
        (/ (/ (fma y (* x 9.0) 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-50) {
		tmp = (fma(9.0, (x * y), b) / z) / c_m;
	} else if (t_1 <= 1e-21) {
		tmp = fma((z * (a * -4.0)), t, b) / (c_m * z);
	} else {
		tmp = (fma(y, (x * 9.0), 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-50)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / z) / c_m);
	elseif (t_1 <= 1e-21)
		tmp = Float64(fma(Float64(z * Float64(a * -4.0)), t, b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(fma(y, Float64(x * 9.0), b) / 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-50], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(N[(N[(z * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * t + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(x * 9.0), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50

   1. Initial program 78.9%

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

      1. associate-/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 74.9%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-lowering-*.f64 72.6

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

   7. Simplified 72.6%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-lowering-*.f64 68.5

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

   7. Applied egg-rr 68.5%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 82.4

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

   7. Simplified 82.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot -4\right), t, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{t\_1}{c\_m \cdot z}\\ \mathbf{elif}\;t\_1 \leq 10^{-21}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

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-50)
      (/ t_1 (* c_m z))
      (if (<= t_1 1e-21)
        (* -4.0 (/ (* t a) c_m))
        (/ (* 9.0 (* x y)) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
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-50) {
		tmp = t_1 / (c_m * z);
	} else if (t_1 <= 1e-21) {
		tmp = -4.0 * ((t * a) / c_m);
	} else {
		tmp = (9.0 * (x * y)) / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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 = y * (x * 9.0d0)
    if (t_1 <= (-5d-50)) then
        tmp = t_1 / (c_m * z)
    else if (t_1 <= 1d-21) then
        tmp = (-4.0d0) * ((t * a) / c_m)
    else
        tmp = (9.0d0 * (x * y)) / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
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 * (x * 9.0);
	double tmp;
	if (t_1 <= -5e-50) {
		tmp = t_1 / (c_m * z);
	} else if (t_1 <= 1e-21) {
		tmp = -4.0 * ((t * a) / c_m);
	} else {
		tmp = (9.0 * (x * y)) / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = y * (x * 9.0)
	tmp = 0
	if t_1 <= -5e-50:
		tmp = t_1 / (c_m * z)
	elif t_1 <= 1e-21:
		tmp = -4.0 * ((t * a) / c_m)
	else:
		tmp = (9.0 * (x * y)) / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
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-50)
		tmp = Float64(t_1 / Float64(c_m * z));
	elseif (t_1 <= 1e-21)
		tmp = Float64(-4.0 * Float64(Float64(t * a) / c_m));
	else
		tmp = Float64(Float64(9.0 * Float64(x * y)) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = y * (x * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e-50)
		tmp = t_1 / (c_m * z);
	elseif (t_1 <= 1e-21)
		tmp = -4.0 * ((t * a) / c_m);
	else
		tmp = (9.0 * (x * y)) / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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-50], N[(t$95$1 / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-21], N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(9.0 * N[(x * y), $MachinePrecision]), $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) < -4.99999999999999968e-50

   1. Initial program 78.9%

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

      1. associate-/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 74.9%

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

   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. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 55.4

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

   7. Simplified 55.4%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

      1. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 77.2%

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

   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 48.5

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

   7. Simplified 48.5%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   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 inf

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

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

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

      2. *-lowering-*.f64 62.5

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

   5. Simplified 62.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 9\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{9 \cdot \left(x \cdot y\right)}{c\_m \cdot z}\\ t_2 := y \cdot \left(x \cdot 9\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-21}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (* 9.0 (* x y)) (* c_m z))) (t_2 (* y (* x 9.0))))
   (*
    c_s
    (if (<= t_2 -5e-50)
      t_1
      (if (<= t_2 1e-21) (* -4.0 (/ (* t a) c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * (x * y)) / (c_m * z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e-50) {
		tmp = t_1;
	} else if (t_2 <= 1e-21) {
		tmp = -4.0 * ((t * a) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * (x * y)) / (c_m * z)
    t_2 = y * (x * 9.0d0)
    if (t_2 <= (-5d-50)) then
        tmp = t_1
    else if (t_2 <= 1d-21) then
        tmp = (-4.0d0) * ((t * a) / c_m)
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * (x * y)) / (c_m * z);
	double t_2 = y * (x * 9.0);
	double tmp;
	if (t_2 <= -5e-50) {
		tmp = t_1;
	} else if (t_2 <= 1e-21) {
		tmp = -4.0 * ((t * a) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = (9.0 * (x * y)) / (c_m * z)
	t_2 = y * (x * 9.0)
	tmp = 0
	if t_2 <= -5e-50:
		tmp = t_1
	elif t_2 <= 1e-21:
		tmp = -4.0 * ((t * a) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(9.0 * Float64(x * y)) / Float64(c_m * z))
	t_2 = Float64(y * Float64(x * 9.0))
	tmp = 0.0
	if (t_2 <= -5e-50)
		tmp = t_1;
	elseif (t_2 <= 1e-21)
		tmp = Float64(-4.0 * Float64(Float64(t * a) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = (9.0 * (x * y)) / (c_m * z);
	t_2 = y * (x * 9.0);
	tmp = 0.0;
	if (t_2 <= -5e-50)
		tmp = t_1;
	elseif (t_2 <= 1e-21)
		tmp = -4.0 * ((t * a) / c_m);
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * 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-50], t$95$1, If[LessEqual[t$95$2, 1e-21], N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999968e-50 or 9.99999999999999908e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. *-lowering-*.f64 59.1

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

   5. Simplified 59.1%

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

   if -4.99999999999999968e-50 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.99999999999999908e-22

   1. Initial program 71.4%

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

      1. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 77.2%

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

   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 48.5

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

   7. Simplified 48.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 9\right) \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \mathbf{elif}\;y \cdot \left(x \cdot 9\right) \leq 10^{-21}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -1.58 \cdot 10^{+32}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t \cdot z}{c\_m \cdot z}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c\_m}\\ \end{array} \end{array} \]

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 (<= a -1.58e+32)
    (* (* a -4.0) (/ (* t z) (* c_m z)))
    (if (<= a 1.65e+161)
      (/ (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 (a <= -1.58e+32) {
		tmp = (a * -4.0) * ((t * z) / (c_m * z));
	} else if (a <= 1.65e+161) {
		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 (a <= -1.58e+32)
		tmp = Float64(Float64(a * -4.0) * Float64(Float64(t * z) / Float64(c_m * z)));
	elseif (a <= 1.65e+161)
		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[a, -1.58e+32], N[(N[(a * -4.0), $MachinePrecision] * N[(N[(t * z), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e+161], 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 a < -1.58000000000000006e32

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 64.2

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

   5. Simplified 64.2%

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

      1. div-inv N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 55.6

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

   7. Applied egg-rr 55.6%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. *-lowering-*.f64 47.5

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

   10. Simplified 47.5%

       \[\leadsto \color{blue}{\left(-4 \cdot \left(a \cdot \left(t \cdot z\right)\right)\right)} \cdot \frac{1}{z \cdot c} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-commutative N/A

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

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

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

       6. div-inv N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 53.6

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

   12. Applied egg-rr 53.6%

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

   if -1.58000000000000006e32 < a < 1.64999999999999999e161

   1. Initial program 80.1%

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

   3. 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 70.2

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

   5. Simplified 70.2%

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

   if 1.64999999999999999e161 < a

   1. Initial program 55.1%

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

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 63.5%

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

   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 58.6

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

   7. Simplified 58.6%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.58 \cdot 10^{+32}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \frac{t \cdot z}{c \cdot z}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \frac{t \cdot a}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* -4.0 (/ (* t a) c_m))))
   (*
    c_s
    (if (<= z -2.15e+56)
      t_1
      (if (<= z 1.6e+112) (/ (fma 9.0 (* x y) b) (* c_m z)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -4.0 * ((t * a) / c_m);
	double tmp;
	if (z <= -2.15e+56) {
		tmp = t_1;
	} else if (z <= 1.6e+112) {
		tmp = fma(9.0, (x * y), b) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(-4.0 * Float64(Float64(t * a) / c_m))
	tmp = 0.0
	if (z <= -2.15e+56)
		tmp = t_1;
	elseif (z <= 1.6e+112)
		tmp = Float64(fma(9.0, Float64(x * y), b) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(-4.0 * N[(N[(t * a), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -2.15e+56], t$95$1, If[LessEqual[z, 1.6e+112], N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1500000000000002e56 or 1.59999999999999993e112 < z

   1. Initial program 55.3%

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

      1. 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. associate-/r/ N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 68.8%

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

   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 53.9

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

   7. Simplified 53.9%

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

   if -2.1500000000000002e56 < z < 1.59999999999999993e112

   1. Initial program 89.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+56}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \frac{t \cdot a}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (<= a -4.5e-131) t_1 (if (<= a 8.6e+60) (/ 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 (a <= -4.5e-131) {
		tmp = t_1;
	} else if (a <= 8.6e+60) {
		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 (a <= (-4.5d-131)) then
        tmp = t_1
    else if (a <= 8.6d+60) 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 (a <= -4.5e-131) {
		tmp = t_1;
	} else if (a <= 8.6e+60) {
		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 a <= -4.5e-131:
		tmp = t_1
	elif a <= 8.6e+60:
		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 (a <= -4.5e-131)
		tmp = t_1;
	elseif (a <= 8.6e+60)
		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 (a <= -4.5e-131)
		tmp = t_1;
	elseif (a <= 8.6e+60)
		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[a, -4.5e-131], t$95$1, If[LessEqual[a, 8.6e+60], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5000000000000002e-131 or 8.59999999999999942e60 < a

   1. Initial program 76.6%

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

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

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

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

   4. Applied egg-rr 72.4%

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

   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 42.9

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

   7. Simplified 42.9%

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

   if -4.5000000000000002e-131 < a < 8.59999999999999942e60

   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 b around inf

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

   5. Simplified 43.6%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-131}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.9% 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 76.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 30.2%

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

6. Final simplification 30.2%

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

Developer Target 1: 80.0% 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 2024198 
(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)))