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

Percentage Accurate: 79.5% → 88.2%

Time: 17.0s

Alternatives: 16

Speedup: 0.9×

• 37.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 8.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{z \cdot \mathsf{fma}\left(a, t \cdot -4, \frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}\right)}{c\_m \cdot z}\\ \mathbf{elif}\;c\_m \leq 7.2 \cdot 10^{+219}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c\_m \cdot z}, \frac{b}{c\_m \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c\_m}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{c\_m}\right)}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 8.8e-21)
    (/ (* z (fma a (* t -4.0) (/ (fma 9.0 (* x y) b) z))) (* c_m z))
    (if (<= c_m 7.2e+219)
      (fma
       a
       (* t (/ -4.0 c_m))
       (fma x (/ (* 9.0 y) (* c_m z)) (/ b (* c_m z))))
      (/ (fma 9.0 (/ (* x y) c_m) (/ (fma a (* -4.0 (* z t)) 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 <= 8.8e-21) {
		tmp = (z * fma(a, (t * -4.0), (fma(9.0, (x * y), b) / z))) / (c_m * z);
	} else if (c_m <= 7.2e+219) {
		tmp = fma(a, (t * (-4.0 / c_m)), fma(x, ((9.0 * y) / (c_m * z)), (b / (c_m * z))));
	} else {
		tmp = fma(9.0, ((x * y) / c_m), (fma(a, (-4.0 * (z * t)), 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 <= 8.8e-21)
		tmp = Float64(Float64(z * fma(a, Float64(t * -4.0), Float64(fma(9.0, Float64(x * y), b) / z))) / Float64(c_m * z));
	elseif (c_m <= 7.2e+219)
		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))));
	else
		tmp = Float64(fma(9.0, Float64(Float64(x * y) / c_m), Float64(fma(a, Float64(-4.0 * Float64(z * t)), 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_] := N[(c$95$s * If[LessEqual[c$95$m, 8.8e-21], N[(N[(z * N[(a * N[(t * -4.0), $MachinePrecision] + N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[c$95$m, 7.2e+219], 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], N[(N[(9.0 * N[(N[(x * y), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if c < 8.8000000000000002e-21

   1. Initial program 86.5%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 8.8000000000000002e-21 < c < 7.20000000000000012e219

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-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. lower-*.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. lower-/.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. Applied rewrites 87.9%

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

   if 7.20000000000000012e219 < c

   1. Initial program 64.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{z \cdot \mathsf{fma}\left(a, t \cdot -4, \frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{z}\right)}{c \cdot z}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+219}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c}, \mathsf{fma}\left(x, \frac{9 \cdot y}{c \cdot z}, \frac{b}{c \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{c}, \frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{c}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := 9 \cdot \left(x \cdot \frac{y}{c\_m \cdot z}\right)\\ t_2 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-183}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{b}{z}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* 9.0 x))))
   (*
    c_s
    (if (<= t_2 -1e+285)
      t_1
      (if (<= t_2 -1e-183)
        (* -4.0 (* a (/ t c_m)))
        (if (<= t_2 2e+127) (/ (/ b z) c_m) t_1))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (c_m * z)));
	double t_2 = y * (9.0 * x);
	double tmp;
	if (t_2 <= -1e+285) {
		tmp = t_1;
	} else if (t_2 <= -1e-183) {
		tmp = -4.0 * (a * (t / c_m));
	} else if (t_2 <= 2e+127) {
		tmp = (b / z) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 9.0d0 * (x * (y / (c_m * z)))
    t_2 = y * (9.0d0 * x)
    if (t_2 <= (-1d+285)) then
        tmp = t_1
    else if (t_2 <= (-1d-183)) then
        tmp = (-4.0d0) * (a * (t / c_m))
    else if (t_2 <= 2d+127) then
        tmp = (b / z) / c_m
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = 9.0 * (x * (y / (c_m * z)));
	double t_2 = y * (9.0 * x);
	double tmp;
	if (t_2 <= -1e+285) {
		tmp = t_1;
	} else if (t_2 <= -1e-183) {
		tmp = -4.0 * (a * (t / c_m));
	} else if (t_2 <= 2e+127) {
		tmp = (b / z) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = 9.0 * (x * (y / (c_m * z)))
	t_2 = y * (9.0 * x)
	tmp = 0
	if t_2 <= -1e+285:
		tmp = t_1
	elif t_2 <= -1e-183:
		tmp = -4.0 * (a * (t / c_m))
	elif t_2 <= 2e+127:
		tmp = (b / z) / c_m
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(9.0 * Float64(x * Float64(y / Float64(c_m * z))))
	t_2 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_2 <= -1e+285)
		tmp = t_1;
	elseif (t_2 <= -1e-183)
		tmp = Float64(-4.0 * Float64(a * Float64(t / c_m)));
	elseif (t_2 <= 2e+127)
		tmp = Float64(Float64(b / z) / c_m);
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = 9.0 * (x * (y / (c_m * z)));
	t_2 = y * (9.0 * x);
	tmp = 0.0;
	if (t_2 <= -1e+285)
		tmp = t_1;
	elseif (t_2 <= -1e-183)
		tmp = -4.0 * (a * (t / c_m));
	elseif (t_2 <= 2e+127)
		tmp = (b / z) / c_m;
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(9.0 * N[(x * N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -1e+285], t$95$1, If[LessEqual[t$95$2, -1e-183], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+127], N[(N[(b / z), $MachinePrecision] / c$95$m), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -9.9999999999999998e284 or 1.99999999999999991e127 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 80.0%

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

   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. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -9.9999999999999998e284 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.00000000000000001e-183

   1. Initial program 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 45.6

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

   7. Applied rewrites 45.6%

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

   if -1.00000000000000001e-183 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.99999999999999991e127

   1. Initial program 85.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 58.9

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

   7. Applied rewrites 58.9%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{+285}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq -1 \cdot 10^{-183}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+165}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot \left(z \cdot a\right), -4, x \cdot \left(9 \cdot y\right)\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

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 (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e+165)
      (/ (* y (/ (* 9.0 x) c_m)) z)
      (if (<= t_1 5e+89)
        (/ (fma -4.0 (* a t) (/ b z)) c_m)
        (/ (fma (* t (* z a)) -4.0 (* x (* 9.0 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 * (9.0 * x);
	double tmp;
	if (t_1 <= -2e+165) {
		tmp = (y * ((9.0 * x) / c_m)) / z;
	} else if (t_1 <= 5e+89) {
		tmp = fma(-4.0, (a * t), (b / z)) / c_m;
	} else {
		tmp = fma((t * (z * a)), -4.0, (x * (9.0 * 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(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e+165)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / c_m)) / z);
	elseif (t_1 <= 5e+89)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c_m);
	else
		tmp = Float64(fma(Float64(t * Float64(z * a)), -4.0, Float64(x * Float64(9.0 * y))) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+165], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+89], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[(t * N[(z * a), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * N[(9.0 * y), $MachinePrecision]), $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) < -1.9999999999999998e165

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      4. lower-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 82.2

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

   10. Applied rewrites 82.2%

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

   if -1.9999999999999998e165 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.99999999999999983e89

   1. Initial program 81.0%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if 4.99999999999999983e89 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 82.1%

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

   3. Taylor expanded in b around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+165}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot \left(z \cdot a\right), -4, x \cdot \left(9 \cdot y\right)\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+165}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e+165)
      (/ (* y (/ (* 9.0 x) c_m)) z)
      (if (<= t_1 1e+63)
        (/ (fma -4.0 (* a t) (/ b z)) c_m)
        (/ (fma x (* 9.0 y) b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -2e+165) {
		tmp = (y * ((9.0 * x) / c_m)) / z;
	} else if (t_1 <= 1e+63) {
		tmp = fma(-4.0, (a * t), (b / z)) / c_m;
	} else {
		tmp = fma(x, (9.0 * y), b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e+165)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / c_m)) / z);
	elseif (t_1 <= 1e+63)
		tmp = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c_m);
	else
		tmp = Float64(fma(x, Float64(9.0 * y), b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+165], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+63], N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1.9999999999999998e165

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      4. lower-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 82.2

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

   10. Applied rewrites 82.2%

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

   if -1.9999999999999998e165 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.00000000000000006e63

   1. Initial program 80.7%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 82.6

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

   8. Applied rewrites 82.6%

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

   if 1.00000000000000006e63 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 74.5

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+165}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e-117)
      (/ (/ (fma 9.0 (* x y) b) c_m) z)
      (if (<= t_1 4e-154)
        (/ (fma (* z (* a t)) -4.0 b) (* c_m z))
        (/ (fma x (* 9.0 y) b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -2e-117) {
		tmp = (fma(9.0, (x * y), b) / c_m) / z;
	} else if (t_1 <= 4e-154) {
		tmp = fma((z * (a * t)), -4.0, b) / (c_m * z);
	} else {
		tmp = fma(x, (9.0 * y), b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e-117)
		tmp = Float64(Float64(fma(9.0, Float64(x * y), b) / c_m) / z);
	elseif (t_1 <= 4e-154)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(c_m * z));
	else
		tmp = Float64(fma(x, Float64(9.0 * y), b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e-117], N[(N[(N[(9.0 * N[(x * y), $MachinePrecision] + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 4e-154], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000006e-117

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      4. lower-*.f64 72.8

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

   7. Applied rewrites 72.8%

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

   if -2.00000000000000006e-117 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.9999999999999999e-154

   1. Initial program 79.4%

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

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

      10. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 77.4

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if 3.9999999999999999e-154 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e+63)
      (/ (* y (/ (* 9.0 x) c_m)) z)
      (if (<= t_1 4e-154)
        (/ (fma (* z (* a t)) -4.0 b) (* c_m z))
        (/ (fma x (* 9.0 y) b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -2e+63) {
		tmp = (y * ((9.0 * x) / c_m)) / z;
	} else if (t_1 <= 4e-154) {
		tmp = fma((z * (a * t)), -4.0, b) / (c_m * z);
	} else {
		tmp = fma(x, (9.0 * y), b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e+63)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / c_m)) / z);
	elseif (t_1 <= 4e-154)
		tmp = Float64(fma(Float64(z * Float64(a * t)), -4.0, b) / Float64(c_m * z));
	else
		tmp = Float64(fma(x, Float64(9.0 * y), b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+63], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 4e-154], N[(N[(N[(z * N[(a * t), $MachinePrecision]), $MachinePrecision] * -4.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000012e63

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      4. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 75.9

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

   10. Applied rewrites 75.9%

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

   if -2.00000000000000012e63 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 3.9999999999999999e-154

   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 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. lower-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. lower-*.f64 N/A

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

      10. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 71.5

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if 3.9999999999999999e-154 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(a \cdot t\right), -4, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := y \cdot \left(9 \cdot x\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c\_m}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* y (* 9.0 x))))
   (*
    c_s
    (if (<= t_1 -2e+63)
      (/ (* y (/ (* 9.0 x) c_m)) z)
      (if (<= t_1 1e-237)
        (/ (fma a (* -4.0 (* z t)) b) (* c_m z))
        (/ (fma x (* 9.0 y) b) (* c_m z)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = y * (9.0 * x);
	double tmp;
	if (t_1 <= -2e+63) {
		tmp = (y * ((9.0 * x) / c_m)) / z;
	} else if (t_1 <= 1e-237) {
		tmp = fma(a, (-4.0 * (z * t)), b) / (c_m * z);
	} else {
		tmp = fma(x, (9.0 * y), b) / (c_m * z);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(y * Float64(9.0 * x))
	tmp = 0.0
	if (t_1 <= -2e+63)
		tmp = Float64(Float64(y * Float64(Float64(9.0 * x) / c_m)) / z);
	elseif (t_1 <= 1e-237)
		tmp = Float64(fma(a, Float64(-4.0 * Float64(z * t)), b) / Float64(c_m * z));
	else
		tmp = Float64(fma(x, Float64(9.0 * y), b) / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, -2e+63], N[(N[(y * N[(N[(9.0 * x), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e-237], N[(N[(a * N[(-4.0 * N[(z * t), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -2.00000000000000012e63

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      4. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 75.9

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

   10. Applied rewrites 75.9%

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

   if -2.00000000000000012e63 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 9.9999999999999999e-238

   1. Initial program 78.4%

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

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

      10. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if 9.9999999999999999e-238 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.7

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

   7. Applied rewrites 73.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(9 \cdot x\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\frac{y \cdot \frac{9 \cdot x}{c}}{z}\\ \mathbf{elif}\;y \cdot \left(9 \cdot x\right) \leq 10^{-237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, -4 \cdot \left(z \cdot t\right), b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (fma -4.0 (* a t) (/ b z)) c_m)))
   (*
    c_s
    (if (<= z -1.1e+51)
      t_1
      (if (<= z 3.1e+140)
        (/ (+ b (- (* y (* 9.0 x)) (* a (* t (* z 4.0))))) (* c_m z))
        t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(-4.0, (a * t), (b / z)) / c_m;
	double tmp;
	if (z <= -1.1e+51) {
		tmp = t_1;
	} else if (z <= 3.1e+140) {
		tmp = (b + ((y * (9.0 * x)) - (a * (t * (z * 4.0))))) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(-4.0, Float64(a * t), Float64(b / z)) / c_m)
	tmp = 0.0
	if (z <= -1.1e+51)
		tmp = t_1;
	elseif (z <= 3.1e+140)
		tmp = Float64(Float64(b + Float64(Float64(y * Float64(9.0 * x)) - Float64(a * Float64(t * Float64(z * 4.0))))) / Float64(c_m * z));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.1e+51], t$95$1, If[LessEqual[z, 3.1e+140], N[(N[(b + N[(N[(y * N[(9.0 * x), $MachinePrecision]), $MachinePrecision] - N[(a * N[(t * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.09999999999999996e51 or 3.1e140 < z

   1. Initial program 52.4%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.0

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

   8. Applied rewrites 81.0%

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

   if -1.09999999999999996e51 < z < 3.1e140

   1. Initial program 94.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{b + \left(y \cdot \left(9 \cdot x\right) - a \cdot \left(t \cdot \left(z \cdot 4\right)\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ (fma -4.0 (* a t) (/ b z)) c_m)))
   (*
    c_s
    (if (<= z -1.1e+51)
      t_1
      (if (<= z 6.6e+210)
        (/ (fma (* z -4.0) (* a 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 = fma(-4.0, (a * t), (b / z)) / c_m;
	double tmp;
	if (z <= -1.1e+51) {
		tmp = t_1;
	} else if (z <= 6.6e+210) {
		tmp = fma((z * -4.0), (a * 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(fma(-4.0, Float64(a * t), Float64(b / z)) / c_m)
	tmp = 0.0
	if (z <= -1.1e+51)
		tmp = t_1;
	elseif (z <= 6.6e+210)
		tmp = Float64(fma(Float64(z * -4.0), Float64(a * 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[(-4.0 * N[(a * t), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -1.1e+51], t$95$1, If[LessEqual[z, 6.6e+210], N[(N[(N[(z * -4.0), $MachinePrecision] * N[(a * t), $MachinePrecision] + 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 < -1.09999999999999996e51 or 6.5999999999999999e210 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 82.5

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

   8. Applied rewrites 82.5%

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

   if -1.09999999999999996e51 < z < 6.5999999999999999e210

   1. Initial program 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 90.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -4, a \cdot t, \mathsf{fma}\left(x, 9 \cdot y, b\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, a \cdot t, \frac{b}{z}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* a (/ t c_m)))))
   (*
    c_s
    (if (<= t -1.65e+216)
      t_1
      (if (<= t 1.2e-59) (/ (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 = -4.0 * (a * (t / c_m));
	double tmp;
	if (t <= -1.65e+216) {
		tmp = t_1;
	} else if (t <= 1.2e-59) {
		tmp = fma(x, (9.0 * y), b) / (c_m * z);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(-4.0 * Float64(a * Float64(t / c_m)))
	tmp = 0.0
	if (t <= -1.65e+216)
		tmp = t_1;
	elseif (t <= 1.2e-59)
		tmp = Float64(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[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -1.65e+216], t$95$1, If[LessEqual[t, 1.2e-59], N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.65e216 or 1.20000000000000008e-59 < t

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -1.65e216 < t < 1.20000000000000008e-59

   1. Initial program 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+216}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* a (/ t c_m)))))
   (*
    c_s
    (if (<= t -1.65e+216)
      t_1
      (if (<= t 1.2e-59) (/ (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 * (a * (t / c_m));
	double tmp;
	if (t <= -1.65e+216) {
		tmp = t_1;
	} else if (t <= 1.2e-59) {
		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(a * Float64(t / c_m)))
	tmp = 0.0
	if (t <= -1.65e+216)
		tmp = t_1;
	elseif (t <= 1.2e-59)
		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[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t, -1.65e+216], t$95$1, If[LessEqual[t, 1.2e-59], 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 t < -1.65e216 or 1.20000000000000008e-59 < t

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -1.65e216 < t < 1.20000000000000008e-59

   1. Initial program 83.3%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+216}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, x \cdot y, b\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{b}{z}}{c\_m}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -6e+30)
    (/ (/ b z) c_m)
    (if (<= b 4.2e+47) (* -4.0 (* a (/ t c_m))) (/ (/ b c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6e+30) {
		tmp = (b / z) / c_m;
	} else if (b <= 4.2e+47) {
		tmp = -4.0 * (a * (t / c_m));
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-6d+30)) then
        tmp = (b / z) / c_m
    else if (b <= 4.2d+47) then
        tmp = (-4.0d0) * (a * (t / c_m))
    else
        tmp = (b / c_m) / z
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6e+30) {
		tmp = (b / z) / c_m;
	} else if (b <= 4.2e+47) {
		tmp = -4.0 * (a * (t / c_m));
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -6e+30:
		tmp = (b / z) / c_m
	elif b <= 4.2e+47:
		tmp = -4.0 * (a * (t / c_m))
	else:
		tmp = (b / c_m) / z
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -6e+30)
		tmp = Float64(Float64(b / z) / c_m);
	elseif (b <= 4.2e+47)
		tmp = Float64(-4.0 * Float64(a * Float64(t / c_m)));
	else
		tmp = Float64(Float64(b / c_m) / z);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -6e+30)
		tmp = (b / z) / c_m;
	elseif (b <= 4.2e+47)
		tmp = -4.0 * (a * (t / c_m));
	else
		tmp = (b / c_m) / z;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -6e+30], N[(N[(b / z), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[b, 4.2e+47], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -5.99999999999999956e30

   1. Initial program 79.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 60.5

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

   7. Applied rewrites 60.5%

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

   if -5.99999999999999956e30 < b < 4.2e47

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 45.5

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

   7. Applied rewrites 45.5%

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

   if 4.2e47 < b

   1. Initial program 82.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 71.8

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

   7. Applied rewrites 71.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -6e+30)
    (/ b (* c_m z))
    (if (<= b 4.2e+47) (* -4.0 (* a (/ t c_m))) (/ (/ b c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6e+30) {
		tmp = b / (c_m * z);
	} else if (b <= 4.2e+47) {
		tmp = -4.0 * (a * (t / c_m));
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: tmp
    if (b <= (-6d+30)) then
        tmp = b / (c_m * z)
    else if (b <= 4.2d+47) then
        tmp = (-4.0d0) * (a * (t / c_m))
    else
        tmp = (b / c_m) / z
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -6e+30) {
		tmp = b / (c_m * z);
	} else if (b <= 4.2e+47) {
		tmp = -4.0 * (a * (t / c_m));
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -6e+30:
		tmp = b / (c_m * z)
	elif b <= 4.2e+47:
		tmp = -4.0 * (a * (t / c_m))
	else:
		tmp = (b / c_m) / z
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -6e+30)
		tmp = Float64(b / Float64(c_m * z));
	elseif (b <= 4.2e+47)
		tmp = Float64(-4.0 * Float64(a * Float64(t / c_m)));
	else
		tmp = Float64(Float64(b / c_m) / z);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -6e+30)
		tmp = b / (c_m * z);
	elseif (b <= 4.2e+47)
		tmp = -4.0 * (a * (t / c_m));
	else
		tmp = (b / c_m) / z;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -6e+30], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e+47], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -5.99999999999999956e30

   1. Initial program 79.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if -5.99999999999999956e30 < b < 4.2e47

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 45.5

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

   7. Applied rewrites 45.5%

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

   if 4.2e47 < b

   1. Initial program 82.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 71.8

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

   7. Applied rewrites 71.8%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \frac{b}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+53}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ b (* c_m z))))
   (*
    c_s
    (if (<= b -6e+30) t_1 (if (<= b 1.55e+53) (* -4.0 (* a (/ t 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 = b / (c_m * z);
	double tmp;
	if (b <= -6e+30) {
		tmp = t_1;
	} else if (b <= 1.55e+53) {
		tmp = -4.0 * (a * (t / c_m));
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m = abs(c)
c\_s = copysign(1.0d0, c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
real(8) function code(c_s, x, y, z, t, a, b, c_m)
    real(8), intent (in) :: c_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b / (c_m * z)
    if (b <= (-6d+30)) then
        tmp = t_1
    else if (b <= 1.55d+53) then
        tmp = (-4.0d0) * (a * (t / 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 = b / (c_m * z);
	double tmp;
	if (b <= -6e+30) {
		tmp = t_1;
	} else if (b <= 1.55e+53) {
		tmp = -4.0 * (a * (t / 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 = b / (c_m * z)
	tmp = 0
	if b <= -6e+30:
		tmp = t_1
	elif b <= 1.55e+53:
		tmp = -4.0 * (a * (t / 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(b / Float64(c_m * z))
	tmp = 0.0
	if (b <= -6e+30)
		tmp = t_1;
	elseif (b <= 1.55e+53)
		tmp = Float64(-4.0 * Float64(a * Float64(t / 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 = b / (c_m * z);
	tmp = 0.0;
	if (b <= -6e+30)
		tmp = t_1;
	elseif (b <= 1.55e+53)
		tmp = -4.0 * (a * (t / 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[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -6e+30], t$95$1, If[LessEqual[b, 1.55e+53], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.99999999999999956e30 or 1.5500000000000001e53 < b

   1. Initial program 80.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -5.99999999999999956e30 < b < 1.5500000000000001e53

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 45.5

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

   7. Applied rewrites 45.5%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+53}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := -4 \cdot \left(t \cdot \frac{a}{c\_m}\right)\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -2.1e-87) t_1 (if (<= a 1.2e+38) (/ 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 <= -2.1e-87) {
		tmp = t_1;
	} else if (a <= 1.2e+38) {
		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 <= (-2.1d-87)) then
        tmp = t_1
    else if (a <= 1.2d+38) 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 <= -2.1e-87) {
		tmp = t_1;
	} else if (a <= 1.2e+38) {
		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 <= -2.1e-87:
		tmp = t_1
	elif a <= 1.2e+38:
		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(t * Float64(a / c_m)))
	tmp = 0.0
	if (a <= -2.1e-87)
		tmp = t_1;
	elseif (a <= 1.2e+38)
		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 <= -2.1e-87)
		tmp = t_1;
	elseif (a <= 1.2e+38)
		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[(t * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -2.1e-87], t$95$1, If[LessEqual[a, 1.2e+38], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.10000000000000007e-87 or 1.20000000000000009e38 < a

   1. Initial program 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 81.0

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.3

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

   9. Applied rewrites 53.3%

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

   if -2.10000000000000007e-87 < a < 1.20000000000000009e38

   1. Initial program 79.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;-4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.3% 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 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 b around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 40.0

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

5. Applied rewrites 40.0%

   \[\leadsto \color{blue}{\frac{b}{z \cdot c}} \]

6. Final simplification 40.0%

   \[\leadsto \frac{b}{c \cdot z} \]
7. Add Preprocessing

Developer Target 1: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024214 
(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)))