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

Percentage Accurate: 91.4% → 94.8%

Time: 6.4s

Alternatives: 7

Speedup: 0.6×

• 28.6% 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: 91.4% 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: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+229} \lor \neg \left(t\_1 \leq 10^{+203}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y \cdot x}{t}, 0.5, -4.5 \cdot z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \end{array} \end{array} \]

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+229) (not (<= t_1 1e+203)))
     (* (/ (fma (/ (* y x) t) 0.5 (* -4.5 z)) a) t)
     (/ (fma y x (* (* -9.0 z) t)) (+ 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+229) || !(t_1 <= 1e+203)) {
		tmp = (fma(((y * x) / t), 0.5, (-4.5 * z)) / a) * t;
	} else {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (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+229) || !(t_1 <= 1e+203))
		tmp = Float64(Float64(fma(Float64(Float64(y * x) / t), 0.5, Float64(-4.5 * z)) / a) * t);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + 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[Or[LessEqual[t$95$1, -5e+229], N[Not[LessEqual[t$95$1, 1e+203]], $MachinePrecision]], N[(N[(N[(N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision] * 0.5 + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000005e229 or 9.9999999999999999e202 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 68.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.9%

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

   if -5.0000000000000005e229 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.9999999999999999e202

   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. 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 97.6

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

   6. Applied rewrites 97.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+229} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 10^{+203}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y \cdot x}{t}, 0.5, -4.5 \cdot z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (* z (/ t a)) -4.5)
     (if (<= t_1 1e+263)
       (/ (fma y x (* (* -9.0 z) t)) (+ a a))
       (* t (* (/ z 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 <= -((double) INFINITY)) {
		tmp = (z * (t / a)) * -4.5;
	} else if (t_1 <= 1e+263) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = t * ((z / 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 <= Float64(-Inf))
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	elseif (t_1 <= 1e+263)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = Float64(t * Float64(Float64(z / a) * -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_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+263], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 50.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 50.7

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000002e263

   1. Initial program 97.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.7

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

   4. Applied rewrites 97.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]
   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 97.7

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

   6. Applied rewrites 97.7%

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

   if 1.00000000000000002e263 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 61.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 95.3%

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

Alternative 3: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-z}{a} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \end{array} \]

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 (<= a 1.5e-24)
   (/ (fma (* t z) -9.0 (* y x)) (* a 2.0))
   (fma (/ (/ y a) 2.0) x (* (/ (- z) a) (* t 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 (a <= 1.5e-24) {
		tmp = fma((t * z), -9.0, (y * x)) / (a * 2.0);
	} else {
		tmp = fma(((y / a) / 2.0), x, ((-z / a) * (t * 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 (a <= 1.5e-24)
		tmp = Float64(fma(Float64(t * z), -9.0, Float64(y * x)) / Float64(a * 2.0));
	else
		tmp = fma(Float64(Float64(y / a) / 2.0), x, Float64(Float64(Float64(-z) / a) * Float64(t * 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[a, 1.5e-24], N[(N[(N[(t * z), $MachinePrecision] * -9.0 + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] / 2.0), $MachinePrecision] * x + N[(N[((-z) / a), $MachinePrecision] * N[(t * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.49999999999999998e-24

   1. Initial program 92.0%

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 92.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 1.49999999999999998e-24 < a

   1. Initial program 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 93.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot z, -9, y \cdot x\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-z}{a} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-47} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \end{array} \]

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-47) (not (<= t_1 2e+19)))
     (* t (* (/ z a) -4.5))
     (/ (* 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-47) || !(t_1 <= 2e+19)) {
		tmp = t * ((z / a) * -4.5);
	} 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-47)) .or. (.not. (t_1 <= 2d+19))) then
        tmp = t * ((z / a) * (-4.5d0))
    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-47) || !(t_1 <= 2e+19)) {
		tmp = t * ((z / a) * -4.5);
	} 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-47) or not (t_1 <= 2e+19):
		tmp = t * ((z / a) * -4.5)
	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-47) || !(t_1 <= 2e+19))
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	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-47) || ~((t_1 <= 2e+19)))
		tmp = t * ((z / a) * -4.5);
	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-47], N[Not[LessEqual[t$95$1, 2e+19]], $MachinePrecision]], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $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) < -5.00000000000000011e-47 or 2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 75.0%

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

   if -5.00000000000000011e-47 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e19

   1. Initial program 96.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.8

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

   4. Applied rewrites 96.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]
   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 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 82.6

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

   9. Applied rewrites 82.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{-47} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+19}\right):\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-61}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -2e-61)
     (* (* z (/ t a)) -4.5)
     (if (<= t_1 2e+19) (/ (* x y) (+ a a)) (* t (* (/ z 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 <= -2e-61) {
		tmp = (z * (t / a)) * -4.5;
	} else if (t_1 <= 2e+19) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t * ((z / 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 <= (-2d-61)) then
        tmp = (z * (t / a)) * (-4.5d0)
    else if (t_1 <= 2d+19) then
        tmp = (x * y) / (a + a)
    else
        tmp = t * ((z / 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 <= -2e-61) {
		tmp = (z * (t / a)) * -4.5;
	} else if (t_1 <= 2e+19) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t * ((z / 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 <= -2e-61:
		tmp = (z * (t / a)) * -4.5
	elif t_1 <= 2e+19:
		tmp = (x * y) / (a + a)
	else:
		tmp = t * ((z / 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 <= -2e-61)
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	elseif (t_1 <= 2e+19)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(t * Float64(Float64(z / 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 <= -2e-61)
		tmp = (z * (t / a)) * -4.5;
	elseif (t_1 <= 2e+19)
		tmp = (x * y) / (a + a);
	else
		tmp = t * ((z / 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, -2e-61], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 2e+19], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2.0000000000000001e-61

   1. Initial program 87.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 75.2%

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

   if -2.0000000000000001e-61 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e19

   1. Initial program 96.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.8

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

   4. Applied rewrites 96.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]
   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 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if 2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 87.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 76.5%

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

Alternative 6: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-27}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

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
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -2e-27)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e+19) (/ (* x y) (+ a a)) (* t (* (/ z 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 <= -2e-27) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e+19) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t * ((z / 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 <= (-2d-27)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d+19) then
        tmp = (x * y) / (a + a)
    else
        tmp = t * ((z / 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 <= -2e-27) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e+19) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t * ((z / 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 <= -2e-27:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e+19:
		tmp = (x * y) / (a + a)
	else:
		tmp = t * ((z / 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 <= -2e-27)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e+19)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(t * Float64(Float64(z / 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 <= -2e-27)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e+19)
		tmp = (x * y) / (a + a);
	else
		tmp = t * ((z / 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, -2e-27], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+19], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2.0000000000000001e-27

   1. Initial program 86.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 74.2%

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

   if -2.0000000000000001e-27 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e19

   1. Initial program 96.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.8

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

   4. Applied rewrites 96.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]
   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 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 82.0

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

   9. Applied rewrites 82.0%

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

   if 2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 87.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 76.5%

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

Alternative 7: 51.4% 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 91.4%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
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 91.8

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

4. Applied rewrites 91.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}}{a \cdot 2} \]
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 91.8

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

6. Applied rewrites 91.8%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 51.7

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

9. Applied rewrites 51.7%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
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 2024338 
(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)))