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

Percentage Accurate: 91.3% → 96.4%

Time: 8.7s

Alternatives: 14

Speedup: 0.7×

• 28.5% 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: 91.3% 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: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+302}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a\_m + a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0))))
   (*
    a_s
    (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+302)))
      (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a_m) x)
      (/ (fma (* z t) -9.0 (* x y)) (+ a_m a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a_m * 2.0);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+302)) {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a_m) * x;
	} else {
		tmp = fma((z * t), -9.0, (x * y)) / (a_m + a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+302))
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a_m) * x);
	else
		tmp = Float64(fma(Float64(z * t), -9.0, Float64(x * y)) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+302]], $MachinePrecision]], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0 or 1.0000000000000001e302 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 78.5%

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

   3. 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 93.7%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 1.0000000000000001e302

   1. Initial program 98.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. +-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 98.4

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

      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 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 98.5

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

   8. Applied rewrites 98.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 10^{+302}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, t \cdot \left(z \cdot \frac{-4.5}{y}\right)\right)}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) (- INFINITY))
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 5e+254)
      (/ (fma (* z t) -9.0 (* x y)) (+ a_m a_m))
      (* (/ (fma 0.5 x (* t (* z (/ -4.5 y)))) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+254) {
		tmp = fma((z * t), -9.0, (x * y)) / (a_m + a_m);
	} else {
		tmp = (fma(0.5, x, (t * (z * (-4.5 / y)))) / a_m) * y;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 5e+254)
		tmp = Float64(fma(Float64(z * t), -9.0, Float64(x * y)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(t * Float64(z * Float64(-4.5 / y)))) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+254], N[(N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x + N[(t * N[(z * N[(-4.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 49.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 99.6%

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

   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 99.6%

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 x y) < 4.99999999999999994e254

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-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 96.2

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

      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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 96.2

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

   8. Applied rewrites 96.2%

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

   if 4.99999999999999994e254 < (*.f64 x y)

   1. Initial program 75.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 100.0%

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

   7. Applied rewrites 100.0%

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

Alternative 3: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a\_m}}{2}, x, \frac{-z}{a\_m} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 5.4e+25)
    (/ (fma (* t -9.0) z (* y x)) (+ a_m a_m))
    (fma (/ (/ y a_m) 2.0) x (* (/ (- z) a_m) (* t 4.5))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 5.4e+25) {
		tmp = fma((t * -9.0), z, (y * x)) / (a_m + a_m);
	} else {
		tmp = fma(((y / a_m) / 2.0), x, ((-z / a_m) * (t * 4.5)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 5.4e+25)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = fma(Float64(Float64(y / a_m) / 2.0), x, Float64(Float64(Float64(-z) / a_m) * Float64(t * 4.5)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 5.4e+25], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a$95$m), $MachinePrecision] / 2.0), $MachinePrecision] * x + N[(N[((-z) / a$95$m), $MachinePrecision] * N[(t * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.4e25

   1. Initial program 94.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. +-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 95.0

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

      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 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 95.0

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

   6. Applied rewrites 95.0%

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

   if 5.4e25 < a

   1. Initial program 80.9%

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

      1. lift-/.f64 N/A

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

      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 91.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{a}}{2}, x, \frac{-z}{a} \cdot \left(t \cdot 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) (- INFINITY))
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 5e+254)
      (/ (fma (* z t) -9.0 (* x y)) (+ a_m a_m))
      (* (/ (* 0.5 x) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+254) {
		tmp = fma((z * t), -9.0, (x * y)) / (a_m + a_m);
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 5e+254)
		tmp = Float64(fma(Float64(z * t), -9.0, Float64(x * y)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(Float64(0.5 * x) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+254], N[(N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 49.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 99.6%

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

   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 99.6%

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 x y) < 4.99999999999999994e254

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-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 96.2

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

      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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 96.2

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

   8. Applied rewrites 96.2%

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

   if 4.99999999999999994e254 < (*.f64 x y)

   1. Initial program 75.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 100.0%

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

   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 96.3%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, y \cdot x\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) (- INFINITY))
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 5e+254)
      (/ (fma (* t -9.0) z (* y x)) (+ a_m a_m))
      (* (/ (* 0.5 x) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+254) {
		tmp = fma((t * -9.0), z, (y * x)) / (a_m + a_m);
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 5e+254)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(y * x)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(Float64(0.5 * x) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+254], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 49.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 99.6%

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

   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 99.6%

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 x y) < 4.99999999999999994e254

   1. Initial program 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-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 96.2

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

      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 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

   if 4.99999999999999994e254 < (*.f64 x y)

   1. Initial program 75.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 100.0%

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

   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 96.3%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -2e+146) (not (<= (* x y) 4e+59)))
    (* (* 0.5 x) (/ y a_m))
    (* (* -4.5 t) (/ z a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = (-4.5 * t) * (z / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (((x * y) <= (-2d+146)) .or. (.not. ((x * y) <= 4d+59))) then
        tmp = (0.5d0 * x) * (y / a_m)
    else
        tmp = ((-4.5d0) * t) * (z / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = (-4.5 * t) * (z / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -2e+146) or not ((x * y) <= 4e+59):
		tmp = (0.5 * x) * (y / a_m)
	else:
		tmp = (-4.5 * t) * (z / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+146) || !(Float64(x * y) <= 4e+59))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	else
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -2e+146) || ~(((x * y) <= 4e+59)))
		tmp = (0.5 * x) * (y / a_m);
	else
		tmp = (-4.5 * t) * (z / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+146], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e+59]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999987e146 or 3.99999999999999989e59 < (*.f64 x y)

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

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

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   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 86.4%

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

   10. Applied rewrites 85.4%

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

   if -1.99999999999999987e146 < (*.f64 x y) < 3.99999999999999989e59

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 69.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a\_m} \cdot -4.5\right) \cdot t\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -2e+146) (not (<= (* x y) 4e+59)))
    (* (* 0.5 x) (/ y a_m))
    (* (* (/ z a_m) -4.5) t))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = ((z / a_m) * -4.5) * t;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (((x * y) <= (-2d+146)) .or. (.not. ((x * y) <= 4d+59))) then
        tmp = (0.5d0 * x) * (y / a_m)
    else
        tmp = ((z / a_m) * (-4.5d0)) * t
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = ((z / a_m) * -4.5) * t;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -2e+146) or not ((x * y) <= 4e+59):
		tmp = (0.5 * x) * (y / a_m)
	else:
		tmp = ((z / a_m) * -4.5) * t
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+146) || !(Float64(x * y) <= 4e+59))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	else
		tmp = Float64(Float64(Float64(z / a_m) * -4.5) * t);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -2e+146) || ~(((x * y) <= 4e+59)))
		tmp = (0.5 * x) * (y / a_m);
	else
		tmp = ((z / a_m) * -4.5) * t;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+146], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e+59]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999987e146 or 3.99999999999999989e59 < (*.f64 x y)

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

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

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   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 86.4%

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

   10. Applied rewrites 85.4%

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

   if -1.99999999999999987e146 < (*.f64 x y) < 3.99999999999999989e59

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 86.6%

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

   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 69.8%

      \[\leadsto \left(\frac{z}{a} \cdot -4.5\right) \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a\_m}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -2e+146) (not (<= (* x y) 4e+59)))
    (* (* 0.5 x) (/ y a_m))
    (* (* z (/ -4.5 a_m)) t))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = (z * (-4.5 / a_m)) * t;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (((x * y) <= (-2d+146)) .or. (.not. ((x * y) <= 4d+59))) then
        tmp = (0.5d0 * x) * (y / a_m)
    else
        tmp = (z * ((-4.5d0) / a_m)) * t
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+146) || !((x * y) <= 4e+59)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = (z * (-4.5 / a_m)) * t;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -2e+146) or not ((x * y) <= 4e+59):
		tmp = (0.5 * x) * (y / a_m)
	else:
		tmp = (z * (-4.5 / a_m)) * t
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+146) || !(Float64(x * y) <= 4e+59))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	else
		tmp = Float64(Float64(z * Float64(-4.5 / a_m)) * t);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -2e+146) || ~(((x * y) <= 4e+59)))
		tmp = (0.5 * x) * (y / a_m);
	else
		tmp = (z * (-4.5 / a_m)) * t;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+146], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e+59]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(-4.5 / a$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999987e146 or 3.99999999999999989e59 < (*.f64 x y)

   1. Initial program 85.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

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

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   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 86.4%

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

   10. Applied rewrites 85.4%

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

   if -1.99999999999999987e146 < (*.f64 x y) < 3.99999999999999989e59

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 86.6%

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

   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 69.8%

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

   10. Applied rewrites 69.8%

       \[\leadsto \left(z \cdot \frac{-4.5}{a}\right) \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+59}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{-4.5}{a}\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+18} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{-19}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4.5 \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -2e+18) (not (<= (* x y) 4e-19)))
    (* (* 0.5 x) (/ y a_m))
    (* z (/ (* -4.5 t) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+18) || !((x * y) <= 4e-19)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = z * ((-4.5 * t) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if (((x * y) <= (-2d+18)) .or. (.not. ((x * y) <= 4d-19))) then
        tmp = (0.5d0 * x) * (y / a_m)
    else
        tmp = z * (((-4.5d0) * t) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -2e+18) || !((x * y) <= 4e-19)) {
		tmp = (0.5 * x) * (y / a_m);
	} else {
		tmp = z * ((-4.5 * t) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -2e+18) or not ((x * y) <= 4e-19):
		tmp = (0.5 * x) * (y / a_m)
	else:
		tmp = z * ((-4.5 * t) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+18) || !(Float64(x * y) <= 4e-19))
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	else
		tmp = Float64(z * Float64(Float64(-4.5 * t) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -2e+18) || ~(((x * y) <= 4e-19)))
		tmp = (0.5 * x) * (y / a_m);
	else
		tmp = z * ((-4.5 * t) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+18], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e-19]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(-4.5 * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e18 or 3.9999999999999999e-19 < (*.f64 x y)

   1. Initial program 87.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 90.8%

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

   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 76.5%

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

   10. Applied rewrites 75.0%

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

   if -2e18 < (*.f64 x y) < 3.9999999999999999e-19

   1. Initial program 95.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 77.8%

      \[\leadsto z \cdot \frac{-4.5 \cdot t}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+18} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{-19}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4.5 \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{-4.5 \cdot \left(z \cdot t\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+18)
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 5e+52) (/ (* -4.5 (* z t)) a_m) (* (/ (* 0.5 x) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+18) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+52) {
		tmp = (-4.5 * (z * t)) / a_m;
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-2d+18)) then
        tmp = (0.5d0 * x) * (y / a_m)
    else if ((x * y) <= 5d+52) then
        tmp = ((-4.5d0) * (z * t)) / a_m
    else
        tmp = ((0.5d0 * x) / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+18) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+52) {
		tmp = (-4.5 * (z * t)) / a_m;
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+18:
		tmp = (0.5 * x) * (y / a_m)
	elif (x * y) <= 5e+52:
		tmp = (-4.5 * (z * t)) / a_m
	else:
		tmp = ((0.5 * x) / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+18)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 5e+52)
		tmp = Float64(Float64(-4.5 * Float64(z * t)) / a_m);
	else
		tmp = Float64(Float64(Float64(0.5 * x) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+18)
		tmp = (0.5 * x) * (y / a_m);
	elseif ((x * y) <= 5e+52)
		tmp = (-4.5 * (z * t)) / a_m;
	else
		tmp = ((0.5 * x) / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+18], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+52], N[(N[(-4.5 * N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e18

   1. Initial program 86.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.2%

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

   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 73.6%

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

   10. Applied rewrites 68.5%

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

   if -2e18 < (*.f64 x y) < 5e52

   1. Initial program 95.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 78.9%

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

   if 5e52 < (*.f64 x y)

   1. Initial program 87.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 89.6%

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

   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 83.4%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{t \cdot z}{a\_m} \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+18)
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 5e+52) (* (/ (* t z) a_m) -4.5) (* (/ (* 0.5 x) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+18) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+52) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-2d+18)) then
        tmp = (0.5d0 * x) * (y / a_m)
    else if ((x * y) <= 5d+52) then
        tmp = ((t * z) / a_m) * (-4.5d0)
    else
        tmp = ((0.5d0 * x) / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+18) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 5e+52) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+18:
		tmp = (0.5 * x) * (y / a_m)
	elif (x * y) <= 5e+52:
		tmp = ((t * z) / a_m) * -4.5
	else:
		tmp = ((0.5 * x) / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+18)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 5e+52)
		tmp = Float64(Float64(Float64(t * z) / a_m) * -4.5);
	else
		tmp = Float64(Float64(Float64(0.5 * x) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+18)
		tmp = (0.5 * x) * (y / a_m);
	elseif ((x * y) <= 5e+52)
		tmp = ((t * z) / a_m) * -4.5;
	else
		tmp = ((0.5 * x) / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+18], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+52], N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e18

   1. Initial program 86.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 96.2%

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

   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 73.6%

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

   10. Applied rewrites 68.5%

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

   if -2e18 < (*.f64 x y) < 5e52

   1. Initial program 95.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 78.7

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

   5. Applied rewrites 78.7%

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

   if 5e52 < (*.f64 x y)

   1. Initial program 87.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 89.6%

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

   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 83.4%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+85}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+146)
    (* (* 0.5 x) (/ y a_m))
    (if (<= (* x y) 4e+85) (* (* -4.5 t) (/ z a_m)) (* (/ (* 0.5 x) a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+146) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 4e+85) {
		tmp = (-4.5 * t) * (z / a_m);
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-2d+146)) then
        tmp = (0.5d0 * x) * (y / a_m)
    else if ((x * y) <= 4d+85) then
        tmp = ((-4.5d0) * t) * (z / a_m)
    else
        tmp = ((0.5d0 * x) / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+146) {
		tmp = (0.5 * x) * (y / a_m);
	} else if ((x * y) <= 4e+85) {
		tmp = (-4.5 * t) * (z / a_m);
	} else {
		tmp = ((0.5 * x) / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+146:
		tmp = (0.5 * x) * (y / a_m)
	elif (x * y) <= 4e+85:
		tmp = (-4.5 * t) * (z / a_m)
	else:
		tmp = ((0.5 * x) / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+146)
		tmp = Float64(Float64(0.5 * x) * Float64(y / a_m));
	elseif (Float64(x * y) <= 4e+85)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a_m));
	else
		tmp = Float64(Float64(Float64(0.5 * x) / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+146)
		tmp = (0.5 * x) * (y / a_m);
	elseif ((x * y) <= 4e+85)
		tmp = (-4.5 * t) * (z / a_m);
	else
		tmp = ((0.5 * x) / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+146], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+85], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999987e146

   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 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 99.7%

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

   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 87.3%

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

   10. Applied rewrites 84.5%

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

   if -1.99999999999999987e146 < (*.f64 x y) < 4.0000000000000001e85

   1. Initial program 95.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 69.6%

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

   if 4.0000000000000001e85 < (*.f64 x y)

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 93.6%

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

   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 87.0%

      \[\leadsto \frac{0.5 \cdot x}{a} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(z \cdot \frac{-4.5 \cdot t}{a\_m}\right) \end{array} \]

FPCore

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

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((-4.5 * t) / a_m));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * (z * (((-4.5d0) * t) / a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((-4.5 * t) / a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (z * ((-4.5 * t) / a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(z * Float64(Float64(-4.5 * t) / a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (z * ((-4.5 * t) / a_m));
end

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 53.4

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

5. Applied rewrites 53.4%

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

7. Applied rewrites 52.5%

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

9. Applied rewrites 52.5%

   \[\leadsto z \cdot \frac{-4.5 \cdot t}{\color{blue}{a}} \]
10. Add Preprocessing

Alternative 14: 50.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(z \cdot \left(\frac{t}{a\_m} \cdot -4.5\right)\right) \end{array} \]

FPCore

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

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((t / a_m) * -4.5));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * (z * ((t / a_m) * (-4.5d0)))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (z * ((t / a_m) * -4.5));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (z * ((t / a_m) * -4.5))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(z * Float64(Float64(t / a_m) * -4.5)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (z * ((t / a_m) * -4.5));
end

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 53.4

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

5. Applied rewrites 53.4%

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

7. Applied rewrites 52.5%

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

Developer Target 1: 94.3% 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 2024339 
(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)))