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

Percentage Accurate: 91.1% → 94.7%

Time: 8.0s

Alternatives: 12

Speedup: 0.7×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a\_m}}{2}, x, \frac{-z}{a\_m} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* a_m 2.0) 5e-61)
    (/ (fma y x (* (* -9.0 z) t)) (+ a_m a_m))
    (fma (/ (/ y a_m) 2.0) x (* (/ (- z) a_m) (* t 4.5))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 5e-61) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a_m + a_m);
	} else {
		tmp = fma(((y / a_m) / 2.0), x, ((-z / a_m) * (t * 4.5)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 5e-61)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a_m + a_m));
	else
		tmp = fma(Float64(Float64(y / a_m) / 2.0), x, Float64(Float64(Float64(-z) / a_m) * Float64(t * 4.5)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 5e-61], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision] * x + N[(N[((-z) / a$95$m), $MachinePrecision] * N[(t * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.9999999999999999e-61

   1. Initial program 90.1%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 90.1

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 90.1

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 90.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 90.1

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

   6. Applied rewrites 90.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.6

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

   8. Applied rewrites 90.6%

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

   if 4.9999999999999999e-61 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 78.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{\color{blue}{a \cdot 2}} \]

      8. times-frac N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{a \cdot 2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a \cdot 2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{y}{a}}}{2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{9 \cdot t}{2}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\color{blue}{\frac{z}{a}}\right) \cdot \frac{9 \cdot t}{2}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \frac{\color{blue}{t \cdot 9}}{2}\right) \]

      22. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

   4. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-z}{a} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+223}\right):\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0))))
   (*
    a_s
    (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+223)))
      (* (* 0.5 y) (/ x a_m))
      (/ (* x y) (+ a_m a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+223)) {
		tmp = (0.5 * y) * (x / a_m);
	} else {
		tmp = (x * y) / (a_m + a_m);
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+223)) {
		tmp = (0.5 * y) * (x / a_m);
	} else {
		tmp = (x * y) / (a_m + a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+223):
		tmp = (0.5 * y) * (x / a_m)
	else:
		tmp = (x * y) / (a_m + a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+223))
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	else
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+223)))
		tmp = (0.5 * y) * (x / a_m);
	else
		tmp = (x * y) / (a_m + a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+223]], $MachinePrecision]], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0 or 1.00000000000000005e223 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 56.2%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 56.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 1.00000000000000005e223

   1. Initial program 99.1%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t\right)}}{a \cdot 2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}\right)}{a \cdot 2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t\right)}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      12. metadata-eval 99.1

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\color{blue}{-9} \cdot z\right) \cdot t\right)}{a \cdot 2} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. lower-*.f64 55.7

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{2 \cdot a}} \]

      3. count-2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

      4. lift-+.f64 55.7

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   9. Applied rewrites 55.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 10^{+223}\right):\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+305} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a\_m} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a\_m + a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (*
    a_s
    (if (or (<= t_1 -1e+305) (not (<= t_1 2e+268)))
      (* (/ (fma (* 0.5 (/ x t)) y (* -4.5 z)) a_m) t)
      (/ (fma (* t z) -9.0 (* y x)) (+ a_m a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -1e+305) || !(t_1 <= 2e+268)) {
		tmp = (fma((0.5 * (x / t)), y, (-4.5 * z)) / a_m) * t;
	} else {
		tmp = fma((t * z), -9.0, (y * x)) / (a_m + a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -1e+305) || !(t_1 <= 2e+268))
		tmp = Float64(Float64(fma(Float64(0.5 * Float64(x / t)), y, Float64(-4.5 * z)) / a_m) * t);
	else
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, -1e+305], N[Not[LessEqual[t$95$1, 2e+268]], $MachinePrecision]], N[(N[(N[(N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision] * y + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999994e304 or 1.9999999999999999e268 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 58.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{\color{blue}{a \cdot 2}} \]

      8. times-frac N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{a \cdot 2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a \cdot 2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{y}{a}}}{2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{9 \cdot t}{2}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\color{blue}{\frac{z}{a}}\right) \cdot \frac{9 \cdot t}{2}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \frac{\color{blue}{t \cdot 9}}{2}\right) \]

      22. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

   4. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x \cdot y}{a \cdot t}\right)} \cdot t \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{-1}{2}} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}}\right) \cdot t \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right)} \cdot t \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right)} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}\right)\right)}\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}}\right)\right)\right) \cdot t \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{z}{a} + \frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)\right)\right)} \cdot t \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)}\right)\right) \cdot t \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)\right)\right) \cdot t} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a} \cdot t} \]

   if -9.9999999999999994e304 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.9999999999999999e268

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+305} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a\_m} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (*
    a_s
    (if (<= t_1 -1e+305)
      (* (/ (fma (* 0.5 (/ x t)) y (* -4.5 z)) a_m) t)
      (if (<= t_1 5e+250)
        (/ (fma (* t z) -9.0 (* y x)) (+ a_m a_m))
        (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a_m) x))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -1e+305) {
		tmp = (fma((0.5 * (x / t)), y, (-4.5 * z)) / a_m) * t;
	} else if (t_1 <= 5e+250) {
		tmp = fma((t * z), -9.0, (y * x)) / (a_m + a_m);
	} else {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a_m) * x;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -1e+305)
		tmp = Float64(Float64(fma(Float64(0.5 * Float64(x / t)), y, Float64(-4.5 * z)) / a_m) * t);
	elseif (t_1 <= 5e+250)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a_m) * x);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -1e+305], N[(N[(N[(N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision] * y + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5e+250], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999994e304

   1. Initial program 63.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{\color{blue}{a \cdot 2}} \]

      8. times-frac N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{a \cdot 2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a \cdot 2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{y}{a}}}{2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{9 \cdot t}{2}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\color{blue}{\frac{z}{a}}\right) \cdot \frac{9 \cdot t}{2}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \frac{\color{blue}{t \cdot 9}}{2}\right) \]

      22. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

   4. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x \cdot y}{a \cdot t}\right)} \cdot t \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{-1}{2}} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}}\right) \cdot t \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right)} \cdot t \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right)} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}\right)\right)}\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}}\right)\right)\right) \cdot t \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{z}{a} + \frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)\right)\right)} \cdot t \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)}\right)\right) \cdot t \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)\right)\right) \cdot t} \]

   7. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a} \cdot t} \]

   if -9.9999999999999994e304 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000002e250

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if 5.0000000000000002e250 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 57.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a\_m} \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a\_m} \cdot y\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (*
    a_s
    (if (<= t_1 -1e+305)
      (* (/ (fma (* 0.5 (/ x t)) y (* -4.5 z)) a_m) t)
      (if (<= t_1 5e+250)
        (/ (fma (* t z) -9.0 (* y x)) (+ a_m a_m))
        (* (/ (fma 0.5 x (* t (* (/ z y) -4.5))) a_m) y))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -1e+305) {
		tmp = (fma((0.5 * (x / t)), y, (-4.5 * z)) / a_m) * t;
	} else if (t_1 <= 5e+250) {
		tmp = fma((t * z), -9.0, (y * x)) / (a_m + a_m);
	} else {
		tmp = (fma(0.5, x, (t * ((z / y) * -4.5))) / a_m) * y;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -1e+305)
		tmp = Float64(Float64(fma(Float64(0.5 * Float64(x / t)), y, Float64(-4.5 * z)) / a_m) * t);
	elseif (t_1 <= 5e+250)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(t * Float64(Float64(z / y) * -4.5))) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -1e+305], N[(N[(N[(N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision] * y + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5e+250], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x + N[(t * N[(N[(z / y), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999994e304

   1. Initial program 63.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{\color{blue}{a \cdot 2}} \]

      8. times-frac N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{a \cdot 2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a \cdot 2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{y}{a}}}{2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{9 \cdot t}{2}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\color{blue}{\frac{z}{a}}\right) \cdot \frac{9 \cdot t}{2}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \frac{\color{blue}{t \cdot 9}}{2}\right) \]

      22. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

   4. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x \cdot y}{a \cdot t}\right)} \cdot t \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{-1}{2}} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} - \color{blue}{\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}}\right) \cdot t \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right)} \cdot t \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{z}{a} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right)} + \left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t}\right)\right) \cdot \frac{-1}{2}\right) \cdot t \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{a \cdot t} \cdot \frac{-1}{2}\right)\right)}\right) \cdot t \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{z}{a}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}}\right)\right)\right) \cdot t \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{z}{a} + \frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)\right)\right)} \cdot t \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)}\right)\right) \cdot t \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{9}{2} \cdot \frac{z}{a}\right)\right)\right) \cdot t} \]

   7. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 \cdot \frac{x}{t}, y, -4.5 \cdot z\right)}{a} \cdot t} \]

   if -9.9999999999999994e304 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000002e250

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if 5.0000000000000002e250 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 57.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} - \color{blue}{\frac{-1}{2}} \cdot \frac{x}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y - \left(\frac{-1}{2} \cdot \frac{x}{a}\right) \cdot y} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right) \cdot y + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right) \cdot y} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot y} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y} + \frac{-1}{2} \cdot \frac{x}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right) \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{x}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot y}\right)\right) \cdot y} \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(\frac{z}{y} \cdot -4.5\right)\right)}{a} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{z}{a\_m} \cdot t\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= (* x y) (- INFINITY))
      t_1
      (if (<= (* x y) -1e-37)
        (/ (* x y) (+ a_m a_m))
        (if (<= (* x y) 2e-6) (* (* (/ z a_m) t) -4.5) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= -1e-37) {
		tmp = (x * y) / (a_m + a_m);
	} else if ((x * y) <= 2e-6) {
		tmp = ((z / a_m) * t) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((x * y) <= -1e-37) {
		tmp = (x * y) / (a_m + a_m);
	} else if ((x * y) <= 2e-6) {
		tmp = ((z / a_m) * t) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (0.5 * y) * (x / a_m)
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = t_1
	elif (x * y) <= -1e-37:
		tmp = (x * y) / (a_m + a_m)
	elif (x * y) <= 2e-6:
		tmp = ((z / a_m) * t) * -4.5
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-37)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (Float64(x * y) <= 2e-6)
		tmp = Float64(Float64(Float64(z / a_m) * t) * -4.5);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = t_1;
	elseif ((x * y) <= -1e-37)
		tmp = (x * y) / (a_m + a_m);
	elseif ((x * y) <= 2e-6)
		tmp = ((z / a_m) * t) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-37], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-6], N[(N[(N[(z / a$95$m), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0 or 1.99999999999999991e-6 < (*.f64 x y)

   1. Initial program 73.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 85.8%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 86.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]

   if -inf.0 < (*.f64 x y) < -1.00000000000000007e-37

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t\right)}}{a \cdot 2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}\right)}{a \cdot 2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t\right)}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      12. metadata-eval 97.6

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\color{blue}{-9} \cdot z\right) \cdot t\right)}{a \cdot 2} \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. lower-*.f64 71.5

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   7. Applied rewrites 71.5%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{2 \cdot a}} \]

      3. count-2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

      4. lift-+.f64 71.5

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   9. Applied rewrites 71.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   if -1.00000000000000007e-37 < (*.f64 x y) < 1.99999999999999991e-6

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.5%

      \[\leadsto \left(\frac{z}{a} \cdot t\right) \cdot -4.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(-4.5 \cdot \frac{z}{a\_m}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= (* x y) (- INFINITY))
      t_1
      (if (<= (* x y) -1e-37)
        (/ (* x y) (+ a_m a_m))
        (if (<= (* x y) 2e-6) (* (* -4.5 (/ z a_m)) t) t_1))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= -1e-37) {
		tmp = (x * y) / (a_m + a_m);
	} else if ((x * y) <= 2e-6) {
		tmp = (-4.5 * (z / a_m)) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (0.5 * y) * (x / a_m);
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((x * y) <= -1e-37) {
		tmp = (x * y) / (a_m + a_m);
	} else if ((x * y) <= 2e-6) {
		tmp = (-4.5 * (z / a_m)) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (0.5 * y) * (x / a_m)
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = t_1
	elif (x * y) <= -1e-37:
		tmp = (x * y) / (a_m + a_m)
	elif (x * y) <= 2e-6:
		tmp = (-4.5 * (z / a_m)) * t
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-37)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (Float64(x * y) <= 2e-6)
		tmp = Float64(Float64(-4.5 * Float64(z / a_m)) * t);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = t_1;
	elseif ((x * y) <= -1e-37)
		tmp = (x * y) / (a_m + a_m);
	elseif ((x * y) <= 2e-6)
		tmp = (-4.5 * (z / a_m)) * t;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-37], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-6], N[(N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0 or 1.99999999999999991e-6 < (*.f64 x y)

   1. Initial program 73.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 85.8%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 86.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]

   if -inf.0 < (*.f64 x y) < -1.00000000000000007e-37

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t\right)}}{a \cdot 2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}\right)}{a \cdot 2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t\right)}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

      12. metadata-eval 97.6

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\color{blue}{-9} \cdot z\right) \cdot t\right)}{a \cdot 2} \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. lower-*.f64 71.5

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   7. Applied rewrites 71.5%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{2 \cdot a}} \]

      3. count-2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

      4. lift-+.f64 71.5

         \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   9. Applied rewrites 71.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   if -1.00000000000000007e-37 < (*.f64 x y) < 1.99999999999999991e-6

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   7. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 40000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{2 \cdot a\_m}, \left(-4.5 \cdot \frac{z}{a\_m}\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* a_m 2.0) 40000000000.0)
    (/ (fma y x (* (* -9.0 z) t)) (+ a_m a_m))
    (fma y (/ x (* 2.0 a_m)) (* (* -4.5 (/ z a_m)) t)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 40000000000.0) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a_m + a_m);
	} else {
		tmp = fma(y, (x / (2.0 * a_m)), ((-4.5 * (z / a_m)) * t));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 40000000000.0)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a_m + a_m));
	else
		tmp = fma(y, Float64(x / Float64(2.0 * a_m)), Float64(Float64(-4.5 * Float64(z / a_m)) * t));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 40000000000.0], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4e10

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 90.6

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 90.6

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 90.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 90.6

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

   6. Applied rewrites 90.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 91.1

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

   8. Applied rewrites 91.1%

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

   if 4e10 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 75.0%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \left(9 \cdot t\right)}{\color{blue}{a \cdot 2}} \]

      8. times-frac N/A

         \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{a \cdot 2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a \cdot 2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{a}}{2}}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{y}{a}}}{2}, x, \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \frac{9 \cdot t}{2}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{9 \cdot t}{2}\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\color{blue}{\frac{z}{a}}\right) \cdot \frac{9 \cdot t}{2}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \frac{\color{blue}{t \cdot 9}}{2}\right) \]

      22. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{a}}{2} \cdot x + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{a}}{2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{a}}{2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{y}{a}}}{2} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{a \cdot 2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{a \cdot 2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} + \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{a \cdot 2}, \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right)\right)} \]

      11. lower-/.f64 95.2

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{x}{a \cdot 2}}, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{a \cdot 2}}, \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{2 \cdot a}}, \left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right)\right) \]

      14. lower-*.f64 95.2

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{\color{blue}{2 \cdot a}}, \left(-\frac{z}{a}\right) \cdot \left(t \cdot 4.5\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{2 \cdot a}, \color{blue}{\left(-\frac{z}{a}\right) \cdot \left(t \cdot \frac{9}{2}\right)}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{2 \cdot a}, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(t \cdot \frac{9}{2}\right)}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{2 \cdot a}, \left(-\frac{z}{a}\right) \cdot \color{blue}{\left(\frac{9}{2} \cdot t\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{x}{2 \cdot a}, \color{blue}{\left(\left(-\frac{z}{a}\right) \cdot \frac{9}{2}\right) \cdot t}\right) \]

   6. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{2 \cdot a}, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a\_m + a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -1e+266) (not (<= (* x y) 2e+273)))
    (* (/ (* 0.5 y) a_m) x)
    (/ (fma (* t z) -9.0 (* y x)) (+ a_m a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -1e+266) || !((x * y) <= 2e+273)) {
		tmp = ((0.5 * y) / a_m) * x;
	} else {
		tmp = fma((t * z), -9.0, (y * x)) / (a_m + a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+266) || !(Float64(x * y) <= 2e+273))
		tmp = Float64(Float64(Float64(0.5 * y) / a_m) * x);
	else
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+266], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+273]], $MachinePrecision]], N[(N[(N[(0.5 * y), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e266 or 1.99999999999999989e273 < (*.f64 x y)

   1. Initial program 61.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 95.8%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]

   if -1e266 < (*.f64 x y) < 1.99999999999999989e273

   1. Initial program 92.9%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 92.9

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 92.9

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 92.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 92.9

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

   6. Applied rewrites 92.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a\_m + a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -1e+266) (not (<= (* x y) 2e+273)))
    (* (/ (* 0.5 y) a_m) x)
    (/ (fma y x (* (* -9.0 z) t)) (+ a_m a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -1e+266) || !((x * y) <= 2e+273)) {
		tmp = ((0.5 * y) / a_m) * x;
	} else {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a_m + a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+266) || !(Float64(x * y) <= 2e+273))
		tmp = Float64(Float64(Float64(0.5 * y) / a_m) * x);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+266], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+273]], $MachinePrecision]], N[(N[(N[(0.5 * y), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e266 or 1.99999999999999989e273 < (*.f64 x y)

   1. Initial program 61.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 95.8%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]

   if -1e266 < (*.f64 x y) < 1.99999999999999989e273

   1. Initial program 92.9%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(\mathsf{neg}\left(z \cdot 9\right)\right)} + x \cdot y}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{t \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(9\right)\right)\right)} + x \cdot y}{a \cdot 2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot \left(\mathsf{neg}\left(9\right)\right)} + x \cdot y}{a \cdot 2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{neg}\left(9\right), x \cdot y\right)}}{a \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{t \cdot z}, \mathsf{neg}\left(9\right), x \cdot y\right)}{a \cdot 2} \]

      11. metadata-eval 92.9

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{x \cdot y}\right)}{a \cdot 2} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lower-*.f64 92.9

          \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   4. Applied rewrites 92.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}}{a \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      3. count-2-rev N/A

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

      4. lower-+.f64 92.9

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

   6. Applied rewrites 92.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 92.9

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

   8. Applied rewrites 92.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{0.5 \cdot y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5 \cdot y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{z}{a\_m} \cdot t\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -1e-37)
    (* (/ (* 0.5 y) a_m) x)
    (if (<= (* x y) 2e-6) (* (* (/ z a_m) t) -4.5) (* (* 0.5 y) (/ x a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-37) {
		tmp = ((0.5 * y) / a_m) * x;
	} else if ((x * y) <= 2e-6) {
		tmp = ((z / a_m) * t) * -4.5;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_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_m
    real(8) :: tmp
    if ((x * y) <= (-1d-37)) then
        tmp = ((0.5d0 * y) / a_m) * x
    else if ((x * y) <= 2d-6) then
        tmp = ((z / a_m) * t) * (-4.5d0)
    else
        tmp = (0.5d0 * y) * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-37) {
		tmp = ((0.5 * y) / a_m) * x;
	} else if ((x * y) <= 2e-6) {
		tmp = ((z / a_m) * t) * -4.5;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -1e-37:
		tmp = ((0.5 * y) / a_m) * x
	elif (x * y) <= 2e-6:
		tmp = ((z / a_m) * t) * -4.5
	else:
		tmp = (0.5 * y) * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -1e-37)
		tmp = Float64(Float64(Float64(0.5 * y) / a_m) * x);
	elseif (Float64(x * y) <= 2e-6)
		tmp = Float64(Float64(Float64(z / a_m) * t) * -4.5);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -1e-37)
		tmp = ((0.5 * y) / a_m) * x;
	elseif ((x * y) <= 2e-6)
		tmp = ((z / a_m) * t) * -4.5;
	else
		tmp = (0.5 * y) * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-37], N[(N[(N[(0.5 * y), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-6], N[(N[(N[(z / a$95$m), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000007e-37

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]

   if -1.00000000000000007e-37 < (*.f64 x y) < 1.99999999999999991e-6

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot z}{a}} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{\color{blue}{t \cdot z}}{a} \cdot -4.5 \]

   5. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.5%

      \[\leadsto \left(\frac{z}{a} \cdot t\right) \cdot -4.5 \]

   if 1.99999999999999991e-6 < (*.f64 x y)

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{1}{2} \cdot \frac{y}{a}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{y}{a}\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} - \color{blue}{\frac{-1}{2}} \cdot \frac{y}{a}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x - \left(\frac{-1}{2} \cdot \frac{y}{a}\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right) \cdot x} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2}\right)\right)} \cdot \frac{t \cdot z}{a \cdot x} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

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

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x} + \frac{-1}{2} \cdot \frac{y}{a}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{y}{a} + \frac{9}{2} \cdot \frac{t \cdot z}{a \cdot x}\right)\right) \cdot x} \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{2} \cdot y}{a} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 83.5%

      \[\leadsto \frac{0.5 \cdot y}{a} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 84.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot \frac{x}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / (a_m + a_m));
}

Fortran

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

Java

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

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((x * y) / (a_m + a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(x * y) / Float64(a_m + a_m)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x} + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}{a \cdot 2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t\right)}}{a \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}\right)}{a \cdot 2} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t\right)}{a \cdot 2} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t\right)}{a \cdot 2} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t\right)}{a \cdot 2} \]

   12. metadata-eval 87.3

       \[\leadsto \frac{\mathsf{fma}\left(y, x, \left(\color{blue}{-9} \cdot z\right) \cdot t\right)}{a \cdot 2} \]

4. Applied rewrites 87.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
6. Step-by-step derivation

   1. lower-*.f64 49.7

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

7. Applied rewrites 49.7%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{a \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{2 \cdot a}} \]

   3. count-2 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

   4. lift-+.f64 49.7

      \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]

9. Applied rewrites 49.7%

   \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]
10. Add Preprocessing

Developer Target 1: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))