Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 83.5% → 97.5%

Time: 12.3s

Alternatives: 12

Speedup: 1.1×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.5% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* (/ (/ y_m (+ z 1.0)) z) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((y_m / (z + 1.0)) / z) * (x_m / z)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (((y_m / (z + 1.0d0)) / z) * (x_m / z)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((y_m / (z + 1.0)) / z) * (x_m / z)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (((y_m / (z + 1.0)) / z) * (x_m / z)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(y_m / Float64(z + 1.0)) / z) * Float64(x_m / z))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (((y_m / (z + 1.0)) / z) * (x_m / z)));
end

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. distribute-lft-in N/A

      \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

   7. *-rgt-identity N/A

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

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

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

   9. /-lowering-/.f64 95.4

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

4. Applied egg-rr 95.4%

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

   1. distribute-lft1-in N/A

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

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

      \[\leadsto \frac{\color{blue}{\frac{y}{z + 1}}}{z} \cdot \frac{x}{z} \]

   5. +-lowering-+.f64 97.5

      \[\leadsto \frac{\frac{y}{\color{blue}{z + 1}}}{z} \cdot \frac{x}{z} \]

6. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]
7. Add Preprocessing

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+118}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ x_m z) (/ y_m (* z z)))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -5e+19)
       t_0
       (if (<= t_1 2e-304)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 4e+118) (* y_m (/ x_m (* z (fma z z z)))) t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / z) * (y_m / (z * z));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_0;
	} else if (t_1 <= 2e-304) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 4e+118) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = t_0;
	elseif (t_1 <= 2e-304)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 4e+118)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -5e+19], t$95$0, If[LessEqual[t$95$1, 2e-304], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+118], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e19 or 3.99999999999999987e118 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 77.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

      7. *-rgt-identity N/A

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

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

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

      9. /-lowering-/.f64 91.2

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 90.7

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

   7. Simplified 90.7%

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

   if -5e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999994e-304

   1. Initial program 68.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.9

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

   5. Simplified 68.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   if 1.99999999999999994e-304 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 3.99999999999999987e118

   1. Initial program 86.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 92.2

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

   4. Applied egg-rr 92.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 4 \cdot 10^{+118}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (fma z z z))))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -5e+19)
       t_0
       (if (<= t_1 4e-39) (* (/ x_m z) (/ y_m z)) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * fma(z, z, z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_0;
	} else if (t_1 <= 4e-39) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = t_0;
	elseif (t_1 <= 4e-39)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -5e+19], t$95$0, If[LessEqual[t$95$1, 4e-39], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e19 or 3.99999999999999972e-39 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 78.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 85.5

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

   4. Applied egg-rr 85.5%

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

   if -5e19 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 3.99999999999999972e-39

   1. Initial program 77.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+220}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) 1e+220)
     (/ (/ x_m z) (/ (fma z z z) y_m))
     (* (/ x_m z) (/ (/ y_m z) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 1e+220) {
		tmp = (x_m / z) / (fma(z, z, z) / y_m);
	} else {
		tmp = (x_m / z) * ((y_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= 1e+220)
		tmp = Float64(Float64(x_m / z) / Float64(fma(z, z, z) / y_m));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 1e+220], N[(N[(x$95$m / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1e220

   1. Initial program 78.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

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

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

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

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

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

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

      9. distribute-lft-in N/A

         \[\leadsto \frac{\frac{x}{z}}{\frac{\color{blue}{z \cdot z + z \cdot 1}}{y}} \]

      10. *-rgt-identity N/A

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

      11. accelerator-lowering-fma.f64 97.5

          \[\leadsto \frac{\frac{x}{z}}{\frac{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}{y}} \]

   4. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]

   if 1e220 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 73.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

      7. *-rgt-identity N/A

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

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

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

      9. /-lowering-/.f64 85.9

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

   4. Applied egg-rr 85.9%

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

      1. distribute-lft1-in N/A

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

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

         \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

         \[\leadsto \frac{\color{blue}{\frac{y}{z + 1}}}{z} \cdot \frac{x}{z} \]

      5. +-lowering-+.f64 95.5

         \[\leadsto \frac{\frac{y}{\color{blue}{z + 1}}}{z} \cdot \frac{x}{z} \]

   6. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

   7. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 95.5

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

   9. Simplified 95.5%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+220}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) 1e+45)
     (* (/ x_m z) (/ y_m (fma z z z)))
     (/ (/ x_m z) (* z (/ z y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 1e+45) {
		tmp = (x_m / z) * (y_m / fma(z, z, z));
	} else {
		tmp = (x_m / z) / (z * (z / y_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= 1e+45)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z * Float64(z / y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 1e+45], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999999993e44

   1. Initial program 78.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

      7. *-rgt-identity N/A

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

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

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

      9. /-lowering-/.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   if 9.9999999999999993e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z + 1}{y}}} \cdot \frac{x}{z \cdot z} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{z + 1}{y}} \cdot \color{blue}{\frac{\frac{x}{z}}{z}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{z}}{\frac{z + 1}{y} \cdot z}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{1}^{-1}} \cdot \frac{x}{z}}{\frac{z + 1}{y} \cdot z} \]

      7. clear-num N/A

         \[\leadsto \frac{{1}^{-1} \cdot \color{blue}{\frac{1}{\frac{z}{x}}}}{\frac{z + 1}{y} \cdot z} \]

      8. inv-pow N/A

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

      9. unpow-prod-down N/A

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

      10. associate-/l* N/A

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

      11. *-lft-identity N/A

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

      12. inv-pow N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{x}}}}{\frac{z + 1}{y} \cdot z} \]

      13. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{z + 1}{y} \cdot z} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z + 1}{y} \cdot z}} \]

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

          \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{z + 1}{y} \cdot z} \]

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

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z + 1}{y} \cdot z}} \]

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

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z + 1}{y}} \cdot z} \]

      18. +-lowering-+.f64 93.9

          \[\leadsto \frac{\frac{x}{z}}{\frac{\color{blue}{z + 1}}{y} \cdot z} \]

   4. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z + 1}{y} \cdot z}} \]

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 93.9

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

   7. Simplified 93.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{+198}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) 2e+198)
     (* (/ x_m z) (/ y_m (fma z z z)))
     (* (/ x_m z) (/ (/ y_m z) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 2e+198) {
		tmp = (x_m / z) * (y_m / fma(z, z, z));
	} else {
		tmp = (x_m / z) * ((y_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= 2e+198)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 2e+198], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000004e198

   1. Initial program 78.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

      7. *-rgt-identity N/A

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

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

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

      9. /-lowering-/.f64 97.5

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

   4. Applied egg-rr 97.5%

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

   if 2.00000000000000004e198 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

      7. *-rgt-identity N/A

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

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

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

      9. /-lowering-/.f64 86.5

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

   4. Applied egg-rr 86.5%

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

      1. distribute-lft1-in N/A

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

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

         \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

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

         \[\leadsto \frac{\color{blue}{\frac{y}{z + 1}}}{z} \cdot \frac{x}{z} \]

      5. +-lowering-+.f64 95.6

         \[\leadsto \frac{\frac{y}{\color{blue}{z + 1}}}{z} \cdot \frac{x}{z} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1}}{z}} \cdot \frac{x}{z} \]

   7. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 95.6

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

   9. Simplified 95.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{+198}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0155:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))))
   (*
    y_s
    (*
     x_s
     (if (<= z -1.0) t_0 (if (<= z 0.0155) (* y_m (/ x_m (* z z))) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.0155) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 0.0155d0) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.0155) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 0.0155:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.0155)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.0155)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 0.0155], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.0155 < z

   1. Initial program 78.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

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

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

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

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 83.4

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

   5. Simplified 83.4%

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

   if -1 < z < 0.0155

   1. Initial program 77.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 76.9

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

   5. Simplified 76.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-/r* N/A

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

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

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

      9. *-lowering-*.f64 80.0

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

   7. Applied egg-rr 80.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;z \leq 0.0155:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m x_m) 5e-224)
     (* (/ x_m z) (/ y_m z))
     (* y_m (/ x_m (* z (fma z z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-224) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-224)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-224], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 4.9999999999999999e-224

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if 4.9999999999999999e-224 < (*.f64 x y)

   1. Initial program 82.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 84.2

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

   4. Applied egg-rr 84.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \left(\frac{x\_m}{z} \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* (/ x_m z) (/ y_m (fma z z z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((x_m / z) * (y_m / fma(z, z, z))));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)))))
end

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. distribute-lft-in N/A

      \[\leadsto \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \cdot \frac{x}{z} \]

   7. *-rgt-identity N/A

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

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

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

   9. /-lowering-/.f64 95.4

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

4. Applied egg-rr 95.4%

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

5. Final simplification 95.4%

   \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]
6. Add Preprocessing

Alternative 10: 93.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \left(\frac{y\_m}{z} \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* (/ y_m z) (/ x_m (fma z z z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((y_m / z) * (x_m / fma(z, z, z))));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(y_m / z) * Float64(x_m / fma(z, z, z)))))
end

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

   7. distribute-lft-in N/A

      \[\leadsto \frac{y}{z} \cdot \frac{x}{\color{blue}{z \cdot z + z \cdot 1}} \]

   8. *-rgt-identity N/A

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

   9. accelerator-lowering-fma.f64 94.7

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

4. Applied egg-rr 94.7%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
5. Add Preprocessing

Alternative 11: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \left(y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* y_m (/ x_m (* z (fma z z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * (x_m / (z * fma(z, z, z)))));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))))))
end

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. accelerator-lowering-fma.f64 81.6

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

4. Applied egg-rr 81.6%

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

5. Final simplification 81.6%

   \[\leadsto y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)} \]
6. Add Preprocessing

Alternative 12: 75.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * (x_m / (z * z))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (y_m * (x_m / (z * z))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * (x_m / (z * z))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m * (x_m / (z * z))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m * Float64(x_m / Float64(z * z)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m * (x_m / (z * z))));
end

Wolfram

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

Derivation

1. Initial program 77.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 66.1

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

5. Simplified 66.1%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

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

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

   7. associate-/r* N/A

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

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

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

   9. *-lowering-*.f64 68.8

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

7. Applied egg-rr 68.8%

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

8. Final simplification 68.8%

   \[\leadsto y \cdot \frac{x}{z \cdot z} \]
9. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

  (/ (* x y) (* (* z z) (+ z 1.0))))