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

Percentage Accurate: 91.1% → 95.3%

Time: 12.5s

Alternatives: 10

Speedup: 0.7×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 100:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m \cdot 2}, x, \frac{t \cdot 4.5}{a\_m} \cdot \left(0 - z\right)\right)\\ \end{array} \end{array} \]

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.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* a_m 2.0) 100.0)
    (/ (fma (* t -9.0) z (* x y)) (* a_m 2.0))
    (fma (/ y (* a_m 2.0)) x (* (/ (* t 4.5) a_m) (- 0.0 z))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 100.0) {
		tmp = fma((t * -9.0), z, (x * y)) / (a_m * 2.0);
	} else {
		tmp = fma((y / (a_m * 2.0)), x, (((t * 4.5) / a_m) * (0.0 - z)));
	}
	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])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 100.0)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(a_m * 2.0));
	else
		tmp = fma(Float64(y / Float64(a_m * 2.0)), x, Float64(Float64(Float64(t * 4.5) / a_m) * Float64(0.0 - z)));
	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.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 100.0], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(t * 4.5), $MachinePrecision] / a$95$m), $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 100

   1. Initial program 95.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   if 100 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 82.9%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \color{blue}{0 - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \color{blue}{0 - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - \frac{\color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2}\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - \color{blue}{z \cdot \frac{9 \cdot t}{a \cdot 2}}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - \color{blue}{z \cdot \frac{9 \cdot t}{a \cdot 2}}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{9 \cdot t}{\color{blue}{2 \cdot a}}\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \color{blue}{\frac{\frac{9 \cdot t}{2}}{a}}\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \color{blue}{\frac{\frac{9 \cdot t}{2}}{a}}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{\frac{\color{blue}{t \cdot 9}}{2}}{a}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{\color{blue}{t \cdot \frac{9}{2}}}{a}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{\color{blue}{t \cdot \frac{9}{2}}}{a}\right) \]

      19. metadata-eval 90.7

          \[\leadsto \mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{t \cdot \color{blue}{4.5}}{a}\right) \]

   4. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, 0 - z \cdot \frac{t \cdot 4.5}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 100:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a \cdot 2}, x, \frac{t \cdot 4.5}{a} \cdot \left(0 - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a\_m}{y}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

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.
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 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 -5e+24)
      (* z (* -4.5 (/ t a_m)))
      (if (<= t_1 2e+44) (/ (* x 0.5) (/ a_m y)) (* t (* -4.5 (/ z a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * 0.5) / (a_m / y);
	} else {
		tmp = t * (-4.5 * (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.
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) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-5d+24)) then
        tmp = z * ((-4.5d0) * (t / a_m))
    else if (t_1 <= 2d+44) then
        tmp = (x * 0.5d0) / (a_m / y)
    else
        tmp = t * ((-4.5d0) * (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;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * 0.5) / (a_m / y);
	} else {
		tmp = t * (-4.5 * (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])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -5e+24:
		tmp = z * (-4.5 * (t / a_m))
	elif t_1 <= 2e+44:
		tmp = (x * 0.5) / (a_m / y)
	else:
		tmp = t * (-4.5 * (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])
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(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+24)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a_m)));
	elseif (t_1 <= 2e+44)
		tmp = Float64(Float64(x * 0.5) / Float64(a_m / y));
	else
		tmp = Float64(t * Float64(-4.5 * 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])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e+24)
		tmp = z * (-4.5 * (t / a_m));
	elseif (t_1 <= 2e+44)
		tmp = (x * 0.5) / (a_m / y);
	else
		tmp = t * (-4.5 * (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.
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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -5e+24], N[(z * N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+44], N[(N[(x * 0.5), $MachinePrecision] / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000045e24

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.5

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

   7. Applied egg-rr 81.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 79.9

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

   9. Applied egg-rr 79.9%

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

   if -5.00000000000000045e24 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000002e44

   1. Initial program 93.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in t around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot \frac{1}{2}} + 0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot \frac{1}{2} + 0 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot \frac{1}{2}\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{a}\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{y}{a}, 0\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2} \cdot y}{a}}, 0\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2} \cdot y}{a}}, 0\right) \]

      9. *-lowering-*.f64 76.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{0.5 \cdot y}}{a}, 0\right) \]

   7. Simplified 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{0.5 \cdot y}{a}, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{2} \cdot y}{a}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{2}}{\frac{a}{y}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{2}}{\frac{a}{y}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot x}}{\frac{a}{y}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot x}}{\frac{a}{y}} \]

      9. /-lowering-/.f64 75.6

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

   9. Applied egg-rr 75.6%

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

   if 2.0000000000000002e44 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.6

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, t \cdot \frac{z}{a\_m}, x \cdot \frac{y}{a\_m \cdot 2}\right)\\ \end{array} \end{array} \]

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.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* a_m 2.0) 1e-14)
    (/ (fma (* t -9.0) z (* x y)) (* a_m 2.0))
    (fma -4.5 (* t (/ z a_m)) (* x (/ y (* a_m 2.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 1e-14) {
		tmp = fma((t * -9.0), z, (x * y)) / (a_m * 2.0);
	} else {
		tmp = fma(-4.5, (t * (z / a_m)), (x * (y / (a_m * 2.0))));
	}
	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])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 1e-14)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(a_m * 2.0));
	else
		tmp = fma(-4.5, Float64(t * Float64(z / a_m)), Float64(x * Float64(y / Float64(a_m * 2.0))));
	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.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 1e-14], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 9.99999999999999999e-15

   1. Initial program 95.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 95.0

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

   4. Applied egg-rr 95.0%

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

   if 9.99999999999999999e-15 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 87.2

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

   5. Simplified 87.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 94.5

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

   7. Applied egg-rr 94.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, t \cdot \frac{z}{a}, x \cdot \frac{y}{a \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

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.
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 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 -5e+24)
      (* z (* -4.5 (/ t a_m)))
      (if (<= t_1 2e+44) (* (* x 0.5) (/ y a_m)) (* t (* -4.5 (/ z a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * 0.5) * (y / a_m);
	} else {
		tmp = t * (-4.5 * (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.
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) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-5d+24)) then
        tmp = z * ((-4.5d0) * (t / a_m))
    else if (t_1 <= 2d+44) then
        tmp = (x * 0.5d0) * (y / a_m)
    else
        tmp = t * ((-4.5d0) * (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;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * 0.5) * (y / a_m);
	} else {
		tmp = t * (-4.5 * (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])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -5e+24:
		tmp = z * (-4.5 * (t / a_m))
	elif t_1 <= 2e+44:
		tmp = (x * 0.5) * (y / a_m)
	else:
		tmp = t * (-4.5 * (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])
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(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+24)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a_m)));
	elseif (t_1 <= 2e+44)
		tmp = Float64(Float64(x * 0.5) * Float64(y / a_m));
	else
		tmp = Float64(t * Float64(-4.5 * 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])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e+24)
		tmp = z * (-4.5 * (t / a_m));
	elseif (t_1 <= 2e+44)
		tmp = (x * 0.5) * (y / a_m);
	else
		tmp = t * (-4.5 * (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.
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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -5e+24], N[(z * N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+44], N[(N[(x * 0.5), $MachinePrecision] * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000045e24

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.5

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

   7. Applied egg-rr 81.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 79.9

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

   9. Applied egg-rr 79.9%

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

   if -5.00000000000000045e24 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000002e44

   1. Initial program 93.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in t around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a} + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot \frac{1}{2}} + 0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot \frac{1}{2} + 0 \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot \frac{1}{2}\right)} + 0 \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{a}\right)} + 0 \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{y}{a}, 0\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2} \cdot y}{a}}, 0\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2} \cdot y}{a}}, 0\right) \]

      9. *-lowering-*.f64 76.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{0.5 \cdot y}}{a}, 0\right) \]

   7. Simplified 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{0.5 \cdot y}{a}, 0\right)} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{2} \cdot y}{a}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 76.3

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

   9. Applied egg-rr 76.3%

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

   if 2.0000000000000002e44 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.6

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

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.
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 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 -5e+24)
      (* z (* -4.5 (/ t a_m)))
      (if (<= t_1 2e+44) (* (* x y) (/ 0.5 a_m)) (* t (* -4.5 (/ z a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * y) * (0.5 / a_m);
	} else {
		tmp = t * (-4.5 * (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.
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) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-5d+24)) then
        tmp = z * ((-4.5d0) * (t / a_m))
    else if (t_1 <= 2d+44) then
        tmp = (x * y) * (0.5d0 / a_m)
    else
        tmp = t * ((-4.5d0) * (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;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+24) {
		tmp = z * (-4.5 * (t / a_m));
	} else if (t_1 <= 2e+44) {
		tmp = (x * y) * (0.5 / a_m);
	} else {
		tmp = t * (-4.5 * (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])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -5e+24:
		tmp = z * (-4.5 * (t / a_m))
	elif t_1 <= 2e+44:
		tmp = (x * y) * (0.5 / a_m)
	else:
		tmp = t * (-4.5 * (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])
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(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+24)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a_m)));
	elseif (t_1 <= 2e+44)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a_m));
	else
		tmp = Float64(t * Float64(-4.5 * 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])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e+24)
		tmp = z * (-4.5 * (t / a_m));
	elseif (t_1 <= 2e+44)
		tmp = (x * y) * (0.5 / a_m);
	else
		tmp = t * (-4.5 * (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.
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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -5e+24], N[(z * N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+44], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000045e24

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.5

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

   7. Applied egg-rr 81.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 79.9

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

   9. Applied egg-rr 79.9%

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

   if -5.00000000000000045e24 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2.0000000000000002e44

   1. Initial program 93.9%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. /-lowering-/.f64 N/A

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

      16. metadata-eval 93.8

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 75.1

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

   7. Simplified 75.1%

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

   if 2.0000000000000002e44 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.6

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+24}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \end{array} \end{array} \]

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.
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 (<= (* t (* z 9.0)) 5e+271)
    (/ (fma (* t -9.0) z (* x y)) (* a_m 2.0))
    (* (/ 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);
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 ((t * (z * 9.0)) <= 5e+271) {
		tmp = fma((t * -9.0), z, (x * y)) / (a_m * 2.0);
	} else {
		tmp = (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])
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(t * Float64(z * 9.0)) <= 5e+271)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(a_m * 2.0));
	else
		tmp = 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.
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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision], 5e+271], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000003e271

   1. Initial program 94.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   if 5.0000000000000003e271 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 73.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 96.7

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

   5. Simplified 96.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 96.8

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 94.2%

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

Alternative 7: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \end{array} \end{array} \]

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.
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 (<= (* t (* z 9.0)) 5e+231)
    (* (fma z (* t -9.0) (* x y)) (/ 0.5 a_m))
    (* (/ 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);
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 ((t * (z * 9.0)) <= 5e+231) {
		tmp = fma(z, (t * -9.0), (x * y)) * (0.5 / a_m);
	} else {
		tmp = (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])
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(t * Float64(z * 9.0)) <= 5e+231)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * Float64(0.5 / a_m));
	else
		tmp = 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.
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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision], 5e+231], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000028e231

   1. Initial program 94.2%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. /-lowering-/.f64 N/A

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

      16. metadata-eval 93.6

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   if 5.00000000000000028e231 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 97.0

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

   5. Simplified 97.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 97.2

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

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.
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) 3.8e-156)
    (* z (* -4.5 (/ t a_m)))
    (* t (* -4.5 (/ z a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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) <= 3.8e-156) {
		tmp = z * (-4.5 * (t / a_m));
	} else {
		tmp = t * (-4.5 * (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.
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) <= 3.8d-156) then
        tmp = z * ((-4.5d0) * (t / a_m))
    else
        tmp = t * ((-4.5d0) * (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;
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) <= 3.8e-156) {
		tmp = z * (-4.5 * (t / a_m));
	} else {
		tmp = t * (-4.5 * (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])
[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) <= 3.8e-156:
		tmp = z * (-4.5 * (t / a_m))
	else:
		tmp = t * (-4.5 * (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])
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) <= 3.8e-156)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a_m)));
	else
		tmp = Float64(t * Float64(-4.5 * 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])){:}
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) <= 3.8e-156)
		tmp = z * (-4.5 * (t / a_m));
	else
		tmp = t * (-4.5 * (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.
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], 3.8e-156], N[(z * N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 3.80000000000000008e-156

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 69.0

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

   5. Simplified 69.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 69.1

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

   7. Applied egg-rr 69.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 63.6

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

   9. Applied egg-rr 63.6%

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

   if 3.80000000000000008e-156 < (*.f64 x y)

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 42.9

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

   5. Simplified 42.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 43.0

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

   7. Applied egg-rr 43.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.8 \cdot 10^{-156}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

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.
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) 1.8e-156)
    (* z (* -4.5 (/ t 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);
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) <= 1.8e-156) {
		tmp = z * (-4.5 * (t / 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.
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) <= 1.8d-156) then
        tmp = z * ((-4.5d0) * (t / 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;
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) <= 1.8e-156) {
		tmp = z * (-4.5 * (t / 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])
[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) <= 1.8e-156:
		tmp = z * (-4.5 * (t / 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])
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) <= 1.8e-156)
		tmp = Float64(z * Float64(-4.5 * Float64(t / a_m)));
	else
		tmp = Float64(-4.5 * Float64(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])){:}
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) <= 1.8e-156)
		tmp = z * (-4.5 * (t / 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.
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], 1.8e-156], N[(z * N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.79999999999999999e-156

   1. Initial program 92.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 69.0

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

   5. Simplified 69.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 69.1

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

   7. Applied egg-rr 69.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 63.6

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

   9. Applied egg-rr 63.6%

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

   if 1.79999999999999999e-156 < (*.f64 x y)

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 42.9

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

   5. Simplified 42.9%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.8 \cdot 10^{-156}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% accurate, 1.6× speedup

Math

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

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.
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 (* -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);
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 * (-4.5 * (t * (z / 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.
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 * ((-4.5d0) * (t * (z / 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;
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 * (-4.5 * (t * (z / 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])
[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 * (-4.5 * (t * (z / 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])
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(-4.5 * Float64(t * Float64(z / 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])){:}
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 * (-4.5 * (t * (z / 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.
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[(-4.5 * N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 57.6

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

5. Simplified 57.6%

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

Developer Target 1: 94.2% 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 2024195 
(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)))