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

Percentage Accurate: 91.1% → 94.4%

Time: 8.3s

Alternatives: 12

Speedup: 0.6×

• 28.4% 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: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;\left(\frac{z}{a} \cdot t\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -9\right) \cdot \frac{t}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -1e+268)
     (* (* (/ z a) t) -4.5)
     (if (<= t_1 2e+189)
       (/ (fma (* t -9.0) z (* y x)) (* a 2.0))
       (* (* z -9.0) (/ t (* 2.0 a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_1 <= 2e+189) {
		tmp = fma((t * -9.0), z, (y * x)) / (a * 2.0);
	} else {
		tmp = (z * -9.0) * (t / (2.0 * a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -1e+268)
		tmp = Float64(Float64(Float64(z / a) * t) * -4.5);
	elseif (t_1 <= 2e+189)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(z * -9.0) * Float64(t / Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+268], N[(N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 2e+189], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * -9.0), $MachinePrecision] * N[(t / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999997e267

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 96.7%

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

   if -9.9999999999999997e267 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e189

   1. Initial program 97.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 2e189 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 78.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 96.3%

      \[\leadsto \left(z \cdot -9\right) \cdot \color{blue}{\frac{t}{2 \cdot a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;\left(\frac{z}{a} \cdot t\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{-52}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;\frac{z \cdot \left(-4.5 \cdot t\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+107}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -1e+268)
     (* (* (/ z a) t) -4.5)
     (if (<= t_1 -5e-66)
       (/ (* -4.5 (* t z)) a)
       (if (<= t_1 1e-52)
         (/ (* x y) (+ a a))
         (if (<= t_1 50000000000000.0)
           (/ (* z (* -4.5 t)) a)
           (if (<= t_1 1e+107)
             (* (* 0.5 x) (/ y a))
             (* (* z (/ t a)) -4.5))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_1 <= -5e-66) {
		tmp = (-4.5 * (t * z)) / a;
	} else if (t_1 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_1 <= 50000000000000.0) {
		tmp = (z * (-4.5 * t)) / a;
	} else if (t_1 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-1d+268)) then
        tmp = ((z / a) * t) * (-4.5d0)
    else if (t_1 <= (-5d-66)) then
        tmp = ((-4.5d0) * (t * z)) / a
    else if (t_1 <= 1d-52) then
        tmp = (x * y) / (a + a)
    else if (t_1 <= 50000000000000.0d0) then
        tmp = (z * ((-4.5d0) * t)) / a
    else if (t_1 <= 1d+107) then
        tmp = (0.5d0 * x) * (y / a)
    else
        tmp = (z * (t / a)) * (-4.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_1 <= -5e-66) {
		tmp = (-4.5 * (t * z)) / a;
	} else if (t_1 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_1 <= 50000000000000.0) {
		tmp = (z * (-4.5 * t)) / a;
	} else if (t_1 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -1e+268:
		tmp = ((z / a) * t) * -4.5
	elif t_1 <= -5e-66:
		tmp = (-4.5 * (t * z)) / a
	elif t_1 <= 1e-52:
		tmp = (x * y) / (a + a)
	elif t_1 <= 50000000000000.0:
		tmp = (z * (-4.5 * t)) / a
	elif t_1 <= 1e+107:
		tmp = (0.5 * x) * (y / a)
	else:
		tmp = (z * (t / a)) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -1e+268)
		tmp = Float64(Float64(Float64(z / a) * t) * -4.5);
	elseif (t_1 <= -5e-66)
		tmp = Float64(Float64(-4.5 * Float64(t * z)) / a);
	elseif (t_1 <= 1e-52)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	elseif (t_1 <= 50000000000000.0)
		tmp = Float64(Float64(z * Float64(-4.5 * t)) / a);
	elseif (t_1 <= 1e+107)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -1e+268)
		tmp = ((z / a) * t) * -4.5;
	elseif (t_1 <= -5e-66)
		tmp = (-4.5 * (t * z)) / a;
	elseif (t_1 <= 1e-52)
		tmp = (x * y) / (a + a);
	elseif (t_1 <= 50000000000000.0)
		tmp = (z * (-4.5 * t)) / a;
	elseif (t_1 <= 1e+107)
		tmp = (0.5 * x) * (y / a);
	else
		tmp = (z * (t / a)) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+268], N[(N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, -5e-66], N[(N[(-4.5 * N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e-52], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(N[(z * N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+107], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999997e267

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 96.7%

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

   if -9.9999999999999997e267 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66

   1. Initial program 98.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 74.0%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e-52

   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 88.4

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

   7. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 88.4

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

   9. Applied rewrites 88.4%

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

   if 1e-52 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5e13

   1. Initial program 99.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 70.3%

      \[\leadsto \left(\frac{z}{a} \cdot t\right) \cdot -4.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 85.9%

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

   if 5e13 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999997e106

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 90.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.6%

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

   10. Applied rewrites 65.9%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 90.4%

      \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{-4.5 \cdot \left(t \cdot z\right)}{a}\\ t_2 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;\left(\frac{z}{a} \cdot t\right) \cdot -4.5\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-52}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{elif}\;t\_2 \leq 50000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+107}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* -4.5 (* t z)) a)) (t_2 (* (* z 9.0) t)))
   (if (<= t_2 -1e+268)
     (* (* (/ z a) t) -4.5)
     (if (<= t_2 -5e-66)
       t_1
       (if (<= t_2 1e-52)
         (/ (* x y) (+ a a))
         (if (<= t_2 50000000000000.0)
           t_1
           (if (<= t_2 1e+107)
             (* (* 0.5 x) (/ y a))
             (* (* z (/ t a)) -4.5))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (-4.5 * (t * z)) / a;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_2 <= -5e-66) {
		tmp = t_1;
	} else if (t_2 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_2 <= 50000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((-4.5d0) * (t * z)) / a
    t_2 = (z * 9.0d0) * t
    if (t_2 <= (-1d+268)) then
        tmp = ((z / a) * t) * (-4.5d0)
    else if (t_2 <= (-5d-66)) then
        tmp = t_1
    else if (t_2 <= 1d-52) then
        tmp = (x * y) / (a + a)
    else if (t_2 <= 50000000000000.0d0) then
        tmp = t_1
    else if (t_2 <= 1d+107) then
        tmp = (0.5d0 * x) * (y / a)
    else
        tmp = (z * (t / a)) * (-4.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (-4.5 * (t * z)) / a;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_2 <= -5e-66) {
		tmp = t_1;
	} else if (t_2 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_2 <= 50000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (-4.5 * (t * z)) / a
	t_2 = (z * 9.0) * t
	tmp = 0
	if t_2 <= -1e+268:
		tmp = ((z / a) * t) * -4.5
	elif t_2 <= -5e-66:
		tmp = t_1
	elif t_2 <= 1e-52:
		tmp = (x * y) / (a + a)
	elif t_2 <= 50000000000000.0:
		tmp = t_1
	elif t_2 <= 1e+107:
		tmp = (0.5 * x) * (y / a)
	else:
		tmp = (z * (t / a)) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-4.5 * Float64(t * z)) / a)
	t_2 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_2 <= -1e+268)
		tmp = Float64(Float64(Float64(z / a) * t) * -4.5);
	elseif (t_2 <= -5e-66)
		tmp = t_1;
	elseif (t_2 <= 1e-52)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	elseif (t_2 <= 50000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+107)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (-4.5 * (t * z)) / a;
	t_2 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_2 <= -1e+268)
		tmp = ((z / a) * t) * -4.5;
	elseif (t_2 <= -5e-66)
		tmp = t_1;
	elseif (t_2 <= 1e-52)
		tmp = (x * y) / (a + a);
	elseif (t_2 <= 50000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+107)
		tmp = (0.5 * x) * (y / a);
	else
		tmp = (z * (t / a)) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(-4.5 * N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+268], N[(N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$2, -5e-66], t$95$1, If[LessEqual[t$95$2, 1e-52], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 50000000000000.0], t$95$1, If[LessEqual[t$95$2, 1e+107], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999997e267

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 96.7%

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

   if -9.9999999999999997e267 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66 or 1e-52 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5e13

   1. Initial program 98.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.8%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e-52

   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 88.4

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

   7. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 88.4

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

   9. Applied rewrites 88.4%

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

   if 5e13 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999997e106

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 90.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.6%

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

   10. Applied rewrites 65.9%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 90.4%

      \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{t \cdot z}{a} \cdot -4.5\\ t_2 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;\left(\frac{z}{a} \cdot t\right) \cdot -4.5\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-52}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{elif}\;t\_2 \leq 50000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+107}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (* t z) a) -4.5)) (t_2 (* (* z 9.0) t)))
   (if (<= t_2 -1e+268)
     (* (* (/ z a) t) -4.5)
     (if (<= t_2 -5e-66)
       t_1
       (if (<= t_2 1e-52)
         (/ (* x y) (+ a a))
         (if (<= t_2 50000000000000.0)
           t_1
           (if (<= t_2 1e+107)
             (* (* 0.5 x) (/ y a))
             (* (* z (/ t a)) -4.5))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t * z) / a) * -4.5;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_2 <= -5e-66) {
		tmp = t_1;
	} else if (t_2 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_2 <= 50000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((t * z) / a) * (-4.5d0)
    t_2 = (z * 9.0d0) * t
    if (t_2 <= (-1d+268)) then
        tmp = ((z / a) * t) * (-4.5d0)
    else if (t_2 <= (-5d-66)) then
        tmp = t_1
    else if (t_2 <= 1d-52) then
        tmp = (x * y) / (a + a)
    else if (t_2 <= 50000000000000.0d0) then
        tmp = t_1
    else if (t_2 <= 1d+107) then
        tmp = (0.5d0 * x) * (y / a)
    else
        tmp = (z * (t / a)) * (-4.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t * z) / a) * -4.5;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_2 <= -5e-66) {
		tmp = t_1;
	} else if (t_2 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else if (t_2 <= 50000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+107) {
		tmp = (0.5 * x) * (y / a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((t * z) / a) * -4.5
	t_2 = (z * 9.0) * t
	tmp = 0
	if t_2 <= -1e+268:
		tmp = ((z / a) * t) * -4.5
	elif t_2 <= -5e-66:
		tmp = t_1
	elif t_2 <= 1e-52:
		tmp = (x * y) / (a + a)
	elif t_2 <= 50000000000000.0:
		tmp = t_1
	elif t_2 <= 1e+107:
		tmp = (0.5 * x) * (y / a)
	else:
		tmp = (z * (t / a)) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t * z) / a) * -4.5)
	t_2 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_2 <= -1e+268)
		tmp = Float64(Float64(Float64(z / a) * t) * -4.5);
	elseif (t_2 <= -5e-66)
		tmp = t_1;
	elseif (t_2 <= 1e-52)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	elseif (t_2 <= 50000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+107)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((t * z) / a) * -4.5;
	t_2 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_2 <= -1e+268)
		tmp = ((z / a) * t) * -4.5;
	elseif (t_2 <= -5e-66)
		tmp = t_1;
	elseif (t_2 <= 1e-52)
		tmp = (x * y) / (a + a);
	elseif (t_2 <= 50000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+107)
		tmp = (0.5 * x) * (y / a);
	else
		tmp = (z * (t / a)) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] * -4.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+268], N[(N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$2, -5e-66], t$95$1, If[LessEqual[t$95$2, 1e-52], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 50000000000000.0], t$95$1, If[LessEqual[t$95$2, 1e+107], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999997e267

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 96.7%

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

   if -9.9999999999999997e267 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66 or 1e-52 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5e13

   1. Initial program 98.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e-52

   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 88.4

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

   7. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 88.4

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

   9. Applied rewrites 88.4%

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

   if 5e13 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999997e106

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 90.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.6%

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

   10. Applied rewrites 65.9%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 90.4%

      \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+271}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-t}{a} \cdot \left(z \cdot 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) 1e+271)
   (/ (fma y x (* (* -9.0 z) t)) (+ a a))
   (fma (/ (/ y a) 2.0) x (* (/ (- t) a) (* z 4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= 1e+271) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = fma(((y / a) / 2.0), x, ((-t / a) * (z * 4.5)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= 1e+271)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = fma(Float64(Float64(y / a) / 2.0), x, Float64(Float64(Float64(-t) / a) * Float64(z * 4.5)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], 1e+271], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] / 2.0), $MachinePrecision] * x + N[(N[((-t) / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 9.99999999999999953e270

   1. Initial program 97.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if 9.99999999999999953e270 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 68.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 97.1

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

   4. Applied rewrites 97.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+271}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-t}{a} \cdot \left(z \cdot 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) 2e+307)
   (/ (fma y x (* (* -9.0 z) t)) (+ a a))
   (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a) x)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= 2e+307) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a) * x;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= 2e+307)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a) * x);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.99999999999999997e307

   1. Initial program 97.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.2

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

   6. Applied rewrites 97.2%

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

   if 1.99999999999999997e307 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 63.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 87.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;\left(\frac{z}{a} \cdot t\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -1e+268)
     (* (* (/ z a) t) -4.5)
     (if (<= t_1 2e+235)
       (/ (fma y x (* (* -9.0 z) t)) (+ a a))
       (* (* (/ z a) -4.5) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -1e+268) {
		tmp = ((z / a) * t) * -4.5;
	} else if (t_1 <= 2e+235) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = ((z / a) * -4.5) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -1e+268)
		tmp = Float64(Float64(Float64(z / a) * t) * -4.5);
	elseif (t_1 <= 2e+235)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(z / a) * -4.5) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+268], N[(N[(N[(z / a), $MachinePrecision] * t), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 2e+235], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.9999999999999997e267

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 96.7%

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

   if -9.9999999999999997e267 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000001e235

   1. Initial program 97.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 2.0000000000000001e235 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 71.3%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

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

Alternative 8: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66} \lor \neg \left(t\_1 \leq 10^{+107}\right):\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -5e-66) (not (<= t_1 1e+107)))
     (* (* z (/ -4.5 a)) t)
     (/ (* x y) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -5e-66) || !(t_1 <= 1e+107)) {
		tmp = (z * (-4.5 / a)) * t;
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-5d-66)) .or. (.not. (t_1 <= 1d+107))) then
        tmp = (z * ((-4.5d0) / a)) * t
    else
        tmp = (x * y) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -5e-66) || !(t_1 <= 1e+107)) {
		tmp = (z * (-4.5 / a)) * t;
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -5e-66) or not (t_1 <= 1e+107):
		tmp = (z * (-4.5 / a)) * t
	else:
		tmp = (x * y) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -5e-66) || !(t_1 <= 1e+107))
		tmp = Float64(Float64(z * Float64(-4.5 / a)) * t);
	else
		tmp = Float64(Float64(x * y) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -5e-66) || ~((t_1 <= 1e+107)))
		tmp = (z * (-4.5 / a)) * t;
	else
		tmp = (x * y) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-66], N[Not[LessEqual[t$95$1, 1e+107]], $MachinePrecision]], N[(N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66 or 9.9999999999999997e106 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.7%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 79.2%

      \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 79.3%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999997e106

   1. Initial program 96.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

   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 77.9

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

   7. Applied rewrites 77.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 77.9

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

   9. Applied rewrites 77.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{-66} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 10^{+107}\right):\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a}\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{-52}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-66)
     (* (* z (/ -4.5 a)) t)
     (if (<= t_1 1e-52) (/ (* x y) (+ a a)) (* (* z (/ t a)) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = (z * (-4.5 / a)) * t;
	} else if (t_1 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-5d-66)) then
        tmp = (z * ((-4.5d0) / a)) * t
    else if (t_1 <= 1d-52) then
        tmp = (x * y) / (a + a)
    else
        tmp = (z * (t / a)) * (-4.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = (z * (-4.5 / a)) * t;
	} else if (t_1 <= 1e-52) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-66:
		tmp = (z * (-4.5 / a)) * t
	elif t_1 <= 1e-52:
		tmp = (x * y) / (a + a)
	else:
		tmp = (z * (t / a)) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = Float64(Float64(z * Float64(-4.5 / a)) * t);
	elseif (t_1 <= 1e-52)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-66)
		tmp = (z * (-4.5 / a)) * t;
	elseif (t_1 <= 1e-52)
		tmp = (x * y) / (a + a);
	else
		tmp = (z * (t / a)) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], N[(N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e-52], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 75.2%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1e-52

   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 88.4

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

   7. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 88.4

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

   9. Applied rewrites 88.4%

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

   if 1e-52 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 67.8%

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

Alternative 10: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a}\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 10^{+107}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e-66)
     (* (* z (/ -4.5 a)) t)
     (if (<= t_1 1e+107) (/ (* x y) (+ a a)) (* (* (/ z a) -4.5) t)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = (z * (-4.5 / a)) * t;
	} else if (t_1 <= 1e+107) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = ((z / a) * -4.5) * t;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-5d-66)) then
        tmp = (z * ((-4.5d0) / a)) * t
    else if (t_1 <= 1d+107) then
        tmp = (x * y) / (a + a)
    else
        tmp = ((z / a) * (-4.5d0)) * t
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = (z * (-4.5 / a)) * t;
	} else if (t_1 <= 1e+107) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = ((z / a) * -4.5) * t;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e-66:
		tmp = (z * (-4.5 / a)) * t
	elif t_1 <= 1e+107:
		tmp = (x * y) / (a + a)
	else:
		tmp = ((z / a) * -4.5) * t
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = Float64(Float64(z * Float64(-4.5 / a)) * t);
	elseif (t_1 <= 1e+107)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(z / a) * -4.5) * t);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e-66)
		tmp = (z * (-4.5 / a)) * t;
	elseif (t_1 <= 1e+107)
		tmp = (x * y) / (a + a);
	else
		tmp = ((z / a) * -4.5) * t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], N[(N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 1e+107], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.99999999999999962e-66

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

      \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 75.2%

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

   if -4.99999999999999962e-66 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999997e106

   1. Initial program 96.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

   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 77.9

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

   7. Applied rewrites 77.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 77.9

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

   9. Applied rewrites 77.9%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.1%

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

   3. Taylor expanded in t around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 88.1%

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

Alternative 11: 52.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) 5e+109)
   (/ (* x y) (+ a a))
   (* (* 0.5 x) (/ y a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= 5e+109) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (((x * y) - ((z * 9.0d0) * t)) <= 5d+109) then
        tmp = (x * y) / (a + a)
    else
        tmp = (0.5d0 * x) * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= 5e+109) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (0.5 * x) * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) - ((z * 9.0) * t)) <= 5e+109:
		tmp = (x * y) / (a + a)
	else:
		tmp = (0.5 * x) * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= 5e+109)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(Float64(0.5 * x) * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) - ((z * 9.0) * t)) <= 5e+109)
		tmp = (x * y) / (a + a);
	else
		tmp = (0.5 * x) * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], 5e+109], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 5.0000000000000001e109

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      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 96.5

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

   4. Applied rewrites 96.5%

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

   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 52.0

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

   7. Applied rewrites 52.0%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   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 52.0

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

   9. Applied rewrites 52.0%

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

   if 5.0000000000000001e109 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 84.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 82.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.5%

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

   10. Applied rewrites 41.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \frac{y}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{x \cdot y}{a + a} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (/ (* x y) (+ a a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (x * y) / (a + a);
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) / (a + a)
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (x * y) / (a + a);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (x * y) / (a + a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(x * y) / Float64(a + a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (x * y) / (a + a);
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.0%

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

   1. lift--.f64 N/A

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

   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 93.5

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

4. Applied rewrites 93.5%

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

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 47.9

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

7. Applied rewrites 47.9%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
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 47.9

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

9. Applied rewrites 47.9%

   \[\leadsto \frac{x \cdot y}{\color{blue}{a + a}} \]
10. Add Preprocessing

Developer Target 1: 93.5% 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 2024320 
(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)))